You're absolutely right. There is no evidence that the Lost Tribes made it to America. Archeology hasn't found proof of the Lamenites, Jaredites, or Nephites. There is no scientific evidence for Mormonism. But of what instance is the lack of scientific evidence? It's not going to change a Mormon's mind - he doesn't believe in Mormonism because of scientific evidence.
The Christian faith is consistent with reason.
It is internally consistent, but rests on fundamental presuppositions. They are ones which I hold too, but you have to recognize your presuppositions and note that they are neither provable nor universally shared.
No evidence of the Tower of Babel, either. The DNA evidence disproves it. The archeological evidence disproves all of the timelines in Bible too. So the faithful find themselves arguing the same argument Mormons make? Curious isn’t it.
Where there should be evidence and there is not evidence this is evidence of falsity. Combine this with actual prophecies that were not fulfilled and you know to a certainty that this was a false prophet. If there were lost tribes of Jews in the Americas we would see architecture, ruins, physical descendents with the genetic markers of Jews, tribes with written languages, linguistic analysis of the indian's languages would match Hebrew or Aramaic, etc., etc.
As far as changing a Mormon's mind, I started this commenting on your belief that you can't show a prophet to be false, that it all depends on "presuppositions." I think I have demonstrated that you can show a prophet to be false (if his prophecies don't come true he is a false prophet) regardless of pressupositions. I don't even see how you can logically argue against the point that if a prophet makes a false prophecy he is a false prophet. As far as whether a person can maintain a faith contrary to reason, I don't know, I've never tried it. I imagine people can, but I think most simply avoid analysis of falsehoods in a cult and leave them be, turning to attack rather than examining the truth of their own faith. Others come to their senses and realize that it's baloney and they are part of an irrational, made up religion and leave.
The thing that has struck me is the amazing truth of the Bible compared to what we know of history from other sources, the proof that is there where it should be, and the coherence and consistency of the Bible. It's a different sort of faith that corresponds to reality . . .