My question is: are these two states identical? If I were perfect, I would presumably be sinless. But if I were sinless, would that in and of itself be enough to be perfect? Or would I just be "not doing some things" but I would need to change other things in my life to meet my redemptive potential?
The word perfect rarely means sinless in the Bible, it generally means well rounded, or mature. The idea of a sinless Christian is foreign to the Bible, except of course those in heaven.
For example Paul uses the words translated into English as perfect here:
Here paul uses the Greek τελειόω teleioō ; to complete, i.e. (literally) accomplish. It means 'to fulfil', 'to carry out' a required course in the sense of a received commission. As this rare context does actually imply actual sinlessness in that he aims for heaven, Paul, as our example avoids possible confusion saying he has in no way achieved it, but aims for it.
In the case mentioned:
The Greek word for perfect is similar, it is τέλειος teleios. Again it means something similar to complete, it means totality or undivided. It does not mean sinless. From the context immediately before this verse we can understand the logic. God is unrestricted in His goodness, so according the disciples of should be 'total' in their love, even loving their enemies. The totality has to do with loving, not just some, but all. This love would never have been presumed to be a sinless, pure love; just a total love in its well rounded and broad application without favouritism.
You can have a house that's halfway through being built. And the work that's done so far can be flawless, with no problems anywhere, but it's still not a perfect house.
Or, likewise, have a look at Adam and Eve. They were created without sin, but when temptation came along, they chose to transgress God's commandments. Jesus, on the other hand, was also without sin, but he stayed that way even though he was tempted in all points, as we are..
This shows the difference between sinlessness and perfection: perfection implies a state of completeness, in which you no longer commit sin because you've matured beyond the point of yielding to temptation.
Jesus's atonement can take away our sins, making up for past wrongdoings in a way that we are incapable of. But when he commands us to be perfect, that calls for effort on our parts as well, to strive to live the Gospel, to work at improving ourselves until it becomes part of our nature.
Everyone has some sins that they just don't understand. Even if they can see how it might tempt some people, the idea of choosing to do it themselves just feels foreign to them. (Meanwhile, everyone has other sins that just feel natural and normal to them.) My understanding of perfection is that, when our faith and humility, repentance and obedience lead us to the point where every sin feels that foreign to us--which will most likely not be during this life--that that's the point at which the perfection that Christ commanded of us has been reached.
I will answer your question of whether being “sinless” is the same as “being perfect” first without dealing with how you further expanded it. I also assume you are taking the English meaning of the word perfect, and not the greek one. This distinction has already been mentioned on other answers.
I think the answer is "no" based on the following examples:
Although he did not commit any sin until he took and ate from the wrong tree, the Bible tells us that he was made "upright": Ecc. 7.29 (NASB, KJV, ASV):
We also see Adam not taking any initiative or decisions of his own until becoming a sinner. Until then, God was leading and he was following. On the other hand, after he became a sinner, he was immediately filled with his own initiative - he began to think he was wise (as we all do today because we were born into his race). After all, he had just eaten from the tree of the knowledge of advantages and disadvantages and was now in full control of the course of his life from his perspective.
So for a season he was sinless but was never perfect.
Ezekiel 28.15-17 plainly states (KJV):
It was because of his perfection that he sinned. Even displaying perfect attitude/dids he couldn't handle being perfect and sinned. His heart was lifted up and he sinned.
So perfect behaviour does not guarantee one will remain sinless.
I would welcome any discussion on the Hebrew term "perfect" used in this passage. But the context seems to indicate it could mean the same as in English.
So no, sinless and perfect seem not to be the same thing.
Now, in light of your question (the rest of it) about how God really wants us to be, consider the types of sinlessness of God the Father, the Son, and the Holy Spirit, and God's ultimate purpose for our lives being (Rom 8.29):
A. God the Father:
"Uncorruptible" "Creator, who is blessed for ever. Amen." - Rom 1.23,25 and the One that "cannot be tempted with evil" - Jas 1.13. and also Jas 1.17 reads (KJV):
He is the only One righteous and no righteousness can be generated apart from Him.
B. God the Son:
Instead of a self-generated righteousness, he always did what the righteous One, the Father, told/revealed him to do. Contrast: Joh 5.30,31 (NASB):
He did what the Father would initiate, to the fullest level:
C. God the Holy Spirit:
Sinless and perfect. In the likeness of God the Son, God the Holy Spirit also acts out of the Father's initiative. Not his own: Joh 16.13-15 (NASB):
The type of righteousness displayed and lived by Jesus is the (realizable) righteousness God wants us believers to receive and experience (never generate) in being each and every day more and more conformed with the image of his Son through a believer experiencing his identification with Jesus in death, burial, and resurrection (Romans 6.1-14, specifically 6.11-13).
THAT kind of living is to be guided by the Holy Spirit and being under grace and not being under law.
Nonetheless, the Christian life is not one of sinless perfection, since we constantly battle our flesh that tries to make us go back and live and walk like Adam did, in sin and death, we deciding what to do next and taking our own initiative.
I hope that helps!