First, thank you very much for posing the questions so well. I am also in the process of re-interpreting the letters of Paul in light of NPP for my own personal theology and spiritual application. Thanks to your reference, I just read the first few chapters of deSilva's Transformation: The Heart of Paul's Gospel, which impressed me as trying to be very faithful to the 1) then current Pharisaic understanding of the Hebrew Bible, 2) Greek word usage, social custom, and the New Testament world in general, 3) Paul's understanding of the ministries of Jesus and the Holy Spirit. He is also sensitive to 1) how Protestant theologies have distorted the meanings of Paul, 2) the variety of approaches taken by Wright and other NPP writers, 3) how his own position makes many Protestants uncomfortable especially because righteousness is not as imputed as in most Protestant theologies.
You are looking "if any NPP proponents could offer some insight into how they individually process the following questions". But why does it have to be NPP, if other Christian traditions can answer them? NPP is only at most 50 years old, while Christians have been struggling with the same questions for 2000 years. The whole point of NPP is to push a big reset button to understand the letters of Paul in a fresh way, especially paying attention to the second temple literature so we are more accurate in understanding the thought world of Paul instead of having reformation era theologians read the wrong things into Paul's words.
The value I most gain from reading NPP is to re-imagine what Jesus and Paul were really teaching and putting myself in the shoes of early Christians living 1st century Greco-Roman culture + Jewish diaspora culture + Messianic expectation. For example, in Chapter 1 last section ("A Great Gift Is a Great Responsibility"), deSilva described the 1st century Roman concept of grace (Gr: charis), which has been butchered in modern times to mean free gift without the recipient needing to reciprocate, while in that period, it was assumed that the recipient has reciprocate as a matter of courtesy. (Beautiful imagery of "three goddesses dancing hand-in-hand in a circle" to show how benefit received should flow among 3 people in a circle).
And who precisely is the NPP proponent you have in mind, especially that there is no single NPP view? How about the Roman Catholic and the Eastern Orthodox traditions, which have been teaching a different model of regeneration and atonement for centuries? I think one area of theology relevant to your question would be whether the right model is imputed righteousness (Lutheran), imparted righteousness (Wesleyan), infused righteousness (Catholic), or deification (Eastern Orthodox). I highly recommend looking into how they would answer your questions as well. Since their positions are a lot more compatible with NPP, do their positions qualify as NPP proponent? I also recommend reading C.S. Lewis who fastidiously tries to steer clear of denominational difference, and as a side byproduct (in my opinion) his views are largely unaffected by NPP. I find it very refreshing to re-read C.S. Lewis after being exposed to NPP. In many places he struggled with your questions in his personal life.
Now to your questions and my two cents from what I think how the Roman Catholic church and C.S. Lewis would answer them:
Mentally speaking, how is the Christian supposed to think of themselves (currently "a child of God" in concept, but not really until the resurrection/judgement of Christ?) and their present relationship to God if they see more failures in their discipleship than victories? Does God's disposition change towards you when you sin or do good?
Even in Judaism, repentance (turning) is the key. DeSilva in chapter 1 adapts Wright's analogy in a wonderful way by likening the Christian trusting Jesus to getting in the car and lets the Holy Spirit drive it as He transforms your heart to be more like Christ. Our responsibility is to NOT get off the car. C.S. Lewis in Narnia: The Voyage of the Dawn Treader has Aslan tear off the dragon skin off Edmund (https://www.desiringgod.org/articles/my-dragon-skin-torn-off). Don't forget many passages in Psalm about the sheep needing to agree to be prodded back to the fold and get back to the right path, implying that our portion is to agree to be led back (sometimes in humble/sheepish acknowledgement of our boo-boo). The point is to agree to remain in the car, the presence of Aslan, the fold. As long as we keep repenting and humble to accept correction, God in his faithfulness will continue to transform us, always welcoming us back like the father of the prodigal son, and always respect our freedom to let God continue to do the painful process of regeneration. I was struck reading Isaiah, Jeremiah, and Ezekiel on how patient and willing to give a second chance God is, even after characterizing Jerusalem as a harlot but still begging her to come back to Him.
How should the Christian deal with the mental pressure presented by the idea that the eternal scales seem to tip with each self-investment (see deSilva's quote above) they make, and not succumb to throwing up their hands in frustration and giving up on pursuing Jesus when they willingly sin in some fashion?
I'm in my late 40s, and I have had my share of doing "self-investment" myself, and sometimes in my clearer moments I see how tall of a mountain I need to climb to kill my bad habits or to exert myself to choose the harder option. Yes, giving up (or hiding) is REAL. But as in those same prophetic books, God is ready to forget our sins when we confess it again (Jer 31:34). NPP, as with C.S. Lewis, seem to teach that we should NOT think in terms of eternal scales (I assume you meant God weighing our bad choices in one side, and right choices in the other side) because then it will be very discouraging trying to pay monthly on a $1 million debt, for example. I think the real measure for the transformation is whether it is getting easier to choose the right course of action. The traditional Roman Catholic practice of going to confession before mass is very therapeutic to get fresh encouragement AS LONG AS we decide to walk to the confession room. As habit becomes easier with practice, I think it's really important to end the rebellion early.
Should the Christian understand the sacrifice of Jesus to cover intentional sins after conversion, or should they see these sins as adding to the "flesh" side of judicial scales weighed upon them at the return/judgement of Christ, and try to do enough good to even off the sin?
The sacrifice of Jesus is a limitless reservoir. But it will only be applied when we ask for it. And it's available post baptism/conversion for sure. It's also clear from deSilva's Transformation book that Judgment day is NOT a weighing of good and bad, but how clean our heart is at that point. In this regards, I think the Roman Catholic idea of venial and mortal sins are spot on. By definition, mortal sin is when we lose your salvation ("we got off the car"). Venial sin is when we are still "willing to be in the car" but has some mud because of the sin. It is VERY IMPORTANT not to equate intention with either mortal / venial. It is far more delicate than this, because Roman Catholic church teaches us that we don't fully know the gravity of our sin, especially sins of weakness. C.S. Lewis has many wonderful ways to say this; at the point where we habitually say "THY will be done" instead of "Thy will be DONE", we have committed a mortal sin, but it's very hard to know at the point of sinning because of this blindness. Usually in retrospect we realize it, run away to confession, then we are in a state of grace again ("get back in the car"). It's NOT about even-ing off the sin, it's about getting back to the program to resume getting cleaned. Of course the more often we get off the car the harder it is to get back in again.
What if when we die we're not clean yet (although we are still in the car)? Then according to Roman Catholic teaching, the car will take us to purgatory and the process continues. While in purgatory, our hopeful eyes are still trained on heaven, helping us to endure the painful process that the Holy Spirit does to cleanse us.
EDIT: ADDITIONAL PROTESTANT-ONLY COMMENT after I finished the Transformation book.
If you're exclusively looking for Protestant NPP solution to your questions, DeSilva seems to be your man as I don't find any trace of Catholic ideas like sacraments, purgatory, asking help from saints, etc. I find possible answers to your questions implied in the book:
Question 1: ... their present relationship to God if they see more failures in their discipleship than victories...
See Chapter 1 - The Necessity of Transformation ... : exegeting Rom 8:13-14 and Gal 5:19-21, and Gal 6:7-10 DeSilva emphasizes that
Paul makes it clear that he is not talking about those who fall perchance into any of these sins, as we all in fact do; he is talking about those who continue in these practices, who make ongoing room to engage in them rather than recognizing them as contrary to God's righteousness and desires for us, and seeking the Spirit's guidance and empowerment to leave those attitudes and practices behind.
So it implies that God's disposition toward us is correlated to not in individual failures, but in how persistent we are to continue in the wrong path.
Question 2: How ... not succumb to throwing up their hands in frustration and giving up on pursuing Jesus ...
See Chapter 2 - God Makes This Transformation Possible ... : deSilva is fairly confident that Rom 7:7-25 have been frequently misread to "mistake [Christians'] ongoing experience of struggle and their occasional failures for the status quo of life in the body ...
The divided self that Paul describes in 7:25 is not the state in which Christ has left the one who trusts him. Rather, it is the state from which God has set the believer free in Christ.
Then we have community support: See Chapter 3 - Paul's Guidance for Living As a Transformed and Transforming Community - Restorative Intervention : using Gal 6:1-2 deSilva shows how the community of believers can be the source of encouragement as members of the "cohort" (my own terminology). Of course, it requires a healthy and safe community so one can be open and receives encouragement not judgment, like Alcoholic Antonymous.
Question 3: ... should they see these sins as adding to the "flesh" side of judicial scales...
See Chapter 2 - You Were Freed for a Fresh Start with God: DeSilva highlights how in God's extraordinary generosity, highlighted in Col 2:13-14:
... as far as God is concerned, there is no baggage in our relationship with him based on what we were. The cross of Jesus is God's solemn declaration that he's not keeping a list in his head of all the things we have done wrong...
CONCLUSION: With the sample treatment by a Protestant NPP scholar, I'm relatively optimistic that other NPP proponent would also answer your questions along a similar vein, AS LONG AS they are daring enough to keep doing semper reformanda. As DeSilva said:
Some theologians will experience discomfort, perhaps even pique, at the nestling of the kind of process usually referred to as "sanctification" within, and as essential to, "justification." In keeping with the commitment of the Reformation church to be semper reformanda, however, it is necessary to consider at least whether we have separated asunder what Paul understood to be joined more integrally and thereby lost sight of important elements of Paul's understanding of God's justifying action on behalf of believers, not to mention the relationship between initial and final justification.