We no get to the real reason that Alma brings up Melchizedek. Alma wants to use Melchizedek as an example, but as an example of a righteous leader with a people who have gone astray. This story is not in the Old Testament, and is not used by Paul. It appears to be information about Melchizedek that was available on the brass plates. We cannot say whether or not Paul's oral tradition preserved this story or not, as it was not part of what Paul used that story to do.
In this case, however, it is precisely the point that Alma wants to make. He is not arguing about priesthood here, for however much the text has similarities to Paul's text in Hebrews. What Alma is doing is arguing for the Atoning Messiah, and for the process of repentance. All of his arguments are to that end, including all of his discussion of priesthood.
In this case, he is once again appealing to the ancient as a model for the present. Melchizedek was a righteous lead with a wicked people. Alma is a righteous leader (and not coincidentally, a priest after the order of Melchizedek) and the Ammonihahites are a wicked people.