You have presented a list of what you call "assumptions". I think it best if we discuss them individually. Let us determine just how much assuming is going on here.
1) that the city described in the BoM was named by locals and not the Lehi group. The inscription on the altars are not Hebrew....so if Nahom was a Hebrew name...then the LDS would have to concede that it was merely a coincidence. LDS apologists point out that the passage in the BoM is in the passive tense. It reads "the place which was called Nahom". Again, the BoM does not specify who called the place Nahom.....given the various interpretations that LDS apologists allow for the BoM in order to avoid anachronisms, it is somewhat odd that they should derive any conclusion from the usage of the passive tense.
Well there is compelling evidence that the city was named by a different group. In each instance where they stopped Nephi told us what they named the location. But in this case at Nahom he says that it was the "place which was called Nahom", a different structure than all previous and following place names given by him. It certainly does indicate that the name of the area was Nahom before they arrived there.
8 And it came to pass that he called the name of the river, Laman, and it emptied into the Red Sea; and the valley was in the borders near the mouth thereof.
9 And when my father saw that the waters of the river emptied into the afountain of the Red Sea, he spake unto Laman, saying: O that thou mightest be like unto this river, continually running into the fountain of all righteousness!
10 And he also spake unto Lemuel: O that thou mightest be like unto this valley, afirm and bsteadfast, and immovable in keeping the commandments of the Lord!
...
14 And it came to pass that my father did speak unto them in the avalley of Lemuel...
6 Now, all these things were said and done as my father dwelt in a tent in the avalley which he called Lemuel.
It is clear that Lehi named these two areas after his sons and made a lesson out of it.
Again, they named it Shazer, whether or not it had an existing name.13 And it came to pass that we traveled for the space of four days, nearly a south-southeast direction, and we did pitch our tents again; and we did call the name of the place aShazer.
So we see that Lehi and his family gave names to all of these places. The text is specific in mentioning that.5 And we did come to the land which we called aBountiful, because of its much fruit and also wild honey; and all these things were prepared of the Lord that we might not perish. And we beheld the sea, which we called Irreantum, which, being interpreted, is many waters.
However in this one case it is different:
This is the only place along their journey that had a name that Nephi didn't mention that they themselves had named. It was the place called Nahom. If it was named by them, he would have kept to the formula and mentioned that. He probably would have mentioned why they named it Nahom, since he had in 4 out of 5 of the others.34 And it came to pass that aIshmael died, and was buried in the place which was called bNahom.
It is a normal thing for an Israelite to rename a place, even if it already has a name. For example, I believe that the Valley of Lamuel surely must have already been discovered by someone before him, and likely had a name that Lehi knew. But in order to teach a lesson to his family he called it the Valley of Lemuel, which Nephi included in the record as a spiritual teaching(the point of the book).
The area where Nahom would have been would indeed have been inhabited by non-hebrew speaking peoples. Nonetheless they would have been semitic speaking. If they had named the region NHM, it doubtlessly would have reminded Lehi of the corresponding NHM in hebrew, which I believe means something like "console or comfort". Being the place where Ishmael was buried, it would appropriate for him to simply not rename the area, since the original already carried a meaninful name.
Couple this with the peculiar evidence that it was "the place which was called" Nahom, and the existence of NHM in southern arabia and we have more than an unlikely assumption here.
Sargon