I don't think anybody can "prove" it. I don't spend a lot of time chasing questions like these, since they are not of paramount importance, but from what I've seen, most Scriptural scholars, both Catholic and Protestant, attest to two James. Could there be a third one? I suppose. But with the information available, nobody can definitively prove anything. Your weakness seems to lie here:
We also know the James of Mat 13:55 is NOT the son of Alphaeus, because Alphaeus had ANOTHER son, and his name was LEVI (also called "Matthew")
Question. Why do you think these two Alphaeus' are one and the same? Alphaeus also was a fairly common name. Levi was a tax collector of all things! If Alphaeus who many believe is the same as the one called Clopas was the father of James, then it is unlikely that Alphaeus the father of Levi is the same person. I've just never seen among the many Scriptural scholars I've read, Protestant and Catholic alike, where they assume these two to be one and the same. In any event, either way, whatever. Not the most important issue in the Church. No Scriptural scholar I've read will come out and definitively state the solution to the "James debate", I guess that is what you would call it. Most concentrate on there being two James.