There's no evidence to corroborate any aspect of the Trojan War legend. Priam has no more going for him in terms of historicity than Gilgamesh does. Some ancient historians take some areas of evidence about the Late Bronze Age that are compatible with the Trojan War legend as licence to believe in a historical war, but (a) that means selectively ignoring the incompatibilities, and (b) that's not what corroboration means.

If Priam existed, and if the Trojan War happened, the answer would still be "unlikely". We have little evidence about literacy and language in Late Bronze Age Wilusa. We know it was incorporated into the Arzawa region as a part of the Hittite Empire in the early 13th century, thanks to the survival of the treaty formalising this arrangement, found at Hattusa; and that suggests some degree of commonality of language. But that tells us nothing about what language(s) were spoken at Wilusa, whether any writing systems were in use there, and if so what they were. The only sample of text that has been found from pre-classical Troy is a seal dating to Troy VIIb2 (late 12th century) with the name of its owner written on it in the Luwian hieroglyphic script. Aside from that, we have one slight indication in a diplomatic letter written at the Arzawan court that the scribe there preferred Hittite to Akkadian (which was the usual diplomatic language in the eastern Mediterranean at the time). Taken together, these points do not bode well for the ability to read Akkadian in 13th-12th century Wilusa.

As a final point, the Standard Version of the Gilgamesh poem wouldn't be composed for another couple of centuries: there were older Babylonian versions floating around, and Hittite too, but evidence on the dissemination of these texts is a bit thin on the ground. That's the Bronze Age for you.