OK, so we're talking about British-Israel and not Mormon teaching. Different cult.
The teaching you are talking about is characteristic of the old Worldwide Church of God (Herbert W. Armstrong's group) and the couple hundred groups that split off from it after HWA died. The WCG was every bit as much a controlling cult as the JW's are. Worse, the teaching is also held dearly by the Identity Movement, which is basically your American Nazis. These guys are utterly racist, and they goose-step around with swastikas and the whole bit. The British Israel teaching is the basis of their racism, since they essentially believe that the white races descended from Israel, and all others are at best inferior, and at worst, subhuman.
Now, I'll be the first to admit that the fact that a doctrine is held by undesirable groups does not make it a false teaching. But this teaching has been pretty well refuted, and I will have to get back to you with some references offering such proof, because I'm supposed to be working right now.
Years ago, I read Armstrong's book, The United States and British Commonwealth in Prophecy, and didn't find it very convincing. As I recall, some of his chief arguments revolved around place names sounding like the names of the tribes, e.g. the Dneiper and Dneister rivers were supposedly named after Dan, who he said traveled that way to England, where London was named after him. The word "British" came, he claimed, from the Hebrew words "berith" (covenant) and "ish" (man), hence "British" = "covenant man" or man with whom God had made a covenant, in other words, Israel. These are truly foolish reasonings, and betray utter ignorance of linguistics and etymology.
The material you cited in your post wasn't any more convincing:
Our proofs have never been properly refuted. The
basic claims we have made are irrefutable. If one cannot deny our
claims and there is nobody else who can it follows that one should
What kind of reasoning is that? Think about it: I believe that there are little purple men with orange polka dots living on the planet Pluto. No one has ever properly refuted my belief. My claim is irrefutable. If you cannot deny my claim and there is nobody else who can, it follows that you should believe it. So do you believe that there are little purple men with orange polka dots living on the planet Pluto?
It isn't always possible to prove a negative (as my example in the previous paragraph illustrates). But the failure to disprove unusual claims does not make them true. The burden of proof lies on the one making the unusual claims to prove that they are true, not on others to refute them.
I haven't looked into this issue in a while, but I will try to do some research on it within the next few days, and get you more information on it. It's going to be a busy weekend (my wife is leaving for a vacation in Alaska), so bear with me.