This "frame challenge" answer may not be what you want, but I hope it's helpful to highlight the fundamental difference between "progress" in theology vs. "progress" in science vs. "progress" in knowing God. Theology prioritizes the "once given" revelation but science thrives on making "novel predictions". In theology, the progress is more unified understanding, and "prediction" / "new knowledge" is frowned upon. But in personal mystical relationship with God, yes, there is "progress" in knowing God more, assisted by greater revelation by God himself (grace of the Holy Spirit) to that individual. But theology to that individual serves merely as a "given" guide for the spiritual journey, rather than something that the individual can "improve" upon. Theology, as in science, is a communal enterprise.
What reason does Christianity offer this skeptical observer not to doubt all denominations simultaneously?
Does the sun and the stars revolve around the earth, or does the earth revolves around the sun? Science finally answered the question, but notice how the conflicting theories didn't make a person skeptical of the existence, the goodness, and the beauty of the sun, the stars, and the earth.
So the reason that Christianity offers one to not doubt all denominations simultaneously is how they are all centered on the love of God who sends his "son" Jesus to "save" us and who eventually will make everything right at the end of time. Denominations may construe how Jesus is "son" differently, may offer different definitions of "save", and may offer different theories of the events leading to the final judgment. But all denominations teach the centrality of the love of God expressed in Jesus with whom we need to unite ourselves by whatever process a denomination prescribes that culminates in Jesus's Second Coming. Please notice how "uniting ourselves with Jesus" (a living Being) is very different than "intellectually adhering to a set of propositions about Jesus." Thus Jesus is the irreducible core of Christianity that no denominations contradicting one another can eliminate: the historical Jesus who truly died, truly resurrected, and who is now glorified "sitting at the right hand of the Father". Without Jesus who is now living, there is no sense in uniting ourselves with Him, and Christianity crumbles from the center.
Naturally we ask who is Jesus, what is His nature? Unfortunately here is where you need to make a choice between Jehovah Witnesses, Latter-day Saints, and mainstream Christianity. Even so, this is a disagreement about our understanding of Jesus, not about Jesus's loving us and making Himself available for us to be united with Him, which all denominations teach as absolutely true. If you're willing to narrow your choice to the 3 mainstream branches of Christianity, the commonality ("hard core tenets") will be more extensive, which is based on the Apostle's Creed and which you can find out from C.S. Lewis's description in Mere Christianity. To them the most important doctrine is Jesus is God (the sine qua non of Christianity), where "God" is understood in the Trinitarian way (Nicene creed) and "Jesus" is understood to be the Incarnation of the 2nd "Person" but who is also 100% human (Chalcedonian definition). It is supremely important for three reasons:
- To preserve monotheism (Jesus is the human being of God, not a separate being)
- Jesus has to be 100% human to save everything in our humanity ("That which is not assumed is not healed"). Jesus also has to be 100% human to be in solidarity with us in facing temptation, but victoriously. And thirdly, Jesus not only shed his glory to be a mere human made out of dust, but also placed himself in solidarity with those of lower socio-economic and cultural status.
- Jesus has to be 100% divine to possess the source of the grace to forgive us, the power to restore our humanity, and eternal life.
Why would it be unreasonable for a skeptical observer to extend this skepticism to all N?
It's not unreasonable, so you can end up being an atheist, a Buddhist, or a Muslim.
But if you think it reasonable that to believe in Jesus is central to your ultimate happiness and to your life after death, the Jesus who forgives and "paid" for your wrongdoings (however they are defined and whatever "paid" means), the Jesus who is God's provision (i.e. the "Lamb of God who takes away the sins of the world"), then despite STILL being skeptical on which denomination has the whole truths about God, this rough idea about Jesus should STILL be good enough to lead you to the right path. In this way, total skepticism is overruled by TRUST in the God who loves us in Jesus.
But if you are skeptical toward the God who sent Jesus in the first place, to the point of not being able to say "yes" to accept the gift, it doesn't matter which denominations you are more skeptical towards, since denominations can only point you in the direction of God (according to their theology). The journey is yours alone. Saying "yes" is when you choose to ride a car or a plane even though you're still skeptical which car brand or which airline is the safest.
Why an outside skeptical observer would have any reason whatsoever to accept these hard core tenets in the first place ...?
Those tenets may not be falsifiable in the same way as in science (since part of the reality that theology deals with is historical and God is not subject to experimentation) but at least they become the basis for seeing the world in a new way which you CAN then "test". C.S. Lewis famously said what is written on his memorial stone in Westminster Abbey:
I believe in Christianity as I believe that the Sun has risen, not only because I see it but because by it I see everything else.
(conclusion of his 1944 paper Is Theology Poetry?)
No potential believer needs to understand the full theology of Jesus to take advantage of Jesus's promise to be the Light to you, just as no one needs to understand planetary movements to benefit from the sun. "Taste and see that the Lord is good; How happy is the person who takes refuge in him!" (Ps 34:8). Faith first, understanding follows. Unlike the Enlightenment prejudice, Christianity teaches that Faith is Rational (listen to the lecture The Light of Reason and the Light of Faith).
Thus, does Christianity share this scientific appreciation for the empirical testability of its claims to any extent whatsoever?
The first thing to be said is to understand what does it mean for theology to be a "science" which Aquinas defines (paraphrased by Fr. Thomas White) to mean "a study of the inner intelligibility of the mystery of God revealed in Christ through the Church." This covers a particular person's knowing the Trinitarian God and His Incarnation through the light of faith strengthened by the grace of the Holy Spirit, which is experiential. Theology understood in this way includes making progress in "knowing God" personally, rather than the modern study of theology which tends to be mostly cerebral. I refer you to the linked video above for a brief introduction of what "empirical testability" means in that context.
But for secondary doctrinal differences among denominations, which are many (just look at this list), the strategy usually has to do with proper interpretation of the Bible and choosing the controlling ideas from a denomination's theology that affect interpretation. I suggest you can make progress to discern a denomination's doctrine's "testability" by investigating it using Analytic Theology for better clarity over the terms used, and for discovering its inner coherence. Meaning, if after trying your best to see a doctrine's coherence there is STILL something that bothers your reason, then it fails your test.
Reading the latest systematic theology from a particular denomination helps too. After comparing various systematic theologies of different denominations and after engaging God in a prayerful manner, I think you can make progress to sense which denomination is "truer" for you.