There's Been Some Amount Of Academic Discussion About Paris' Two Names - Usually In Terms Of Which Is

There's been some amount of academic discussion about Paris' two names - usually in terms of which is earlier and where they come from and what epithets are used with which name. (Most of his epithets "belong" only to Alexander, if you're curious.) But, a small branch of it is "who uses what name, in-story, in the Iliad" ; Ann Suter (this woman, uhh her ideas are pretty crazy so approach with awareness of that), I.F. de Jong and, commenting on especially the latter's article, Michael Lloyd.

I lean more towards Lloyd's assessment that de Jong's premise (that "Paris" is between the Trojans and "Alexander" for the Achaeans as a sort of 'international' name) can't really be supported. But! That doesn't mean you still can't have fun with the split in names and get something in terms of character and worldbuilding out of that!

So, first of all, in the Iliad, "Alexander" is used far more than "Paris". Only Hektor ever uses "Paris" in direct speech, about or to him (we'll get back to Hektor).

Everyone else, Achaean or Trojan, uses Alexander.

Both Suter and de Jong would, in various ways, either ignore this or explain it away as a "this only happens when the Trojans are talking to Achaeans" (Hektor, before the duel), or "what is said is going to be said to Achaeans" (Priam, telling Idaeus what to report to the Achaean commanders). Honestly, that seems overly complicated and not very reasonable to me. Especially in the case of Priam, if Paris was the name he's most used to using, there is no reason for him not to use Paris and then Idaeus simply switches when reporting the speech to the Achaeans. Yes, reported speech/instructions are usually relayed verbatim, but switching a name wouldn't be changing what's actually been said.

And, anyway, coming back to Hektor, who is the one to most consistently use Paris? Also uses Alexander, when thinking to himself, in his own head. (He also uses Paris to Achilles.)

Myth-wise, in various later sources you get the very logical conclusion of "one name was given by his foster father, the other by his royal parents". (Though there's not necessarily any consistency, even with one writer, which name was given by whom.)

Given the way the Iliad prioritises Alexander, I'd go with that Alexander is the name Priam and Hecuba gave their son, even if he was going to be exposed, before giving him away. Given how Alexander is used by basically everyone to address him, this would make good sense, I think. The Achaeans would only know of Alexander, prince of Troy, and that is certainly the name most/all Trojans would use. Paris is then the name given him by his foster father. Hektor using it can be turned into a look into their relationship, because what you see is Hektor using the name of the "outsider" (by a bare technically "not" his brother), to insult his brother, when he's angry. A verbal distance to add to the emotional one, if not one that's complete and sometimes blurs.

(This doesn't take into account post-Iliad sources, where 'Paris' vastly outnumber the uses of 'Alexander'.)