Interestingly we have surviving Hittite letters complaining about the Achaeans causing some trouble in the vicinity of Troy along with a guy whose name sounds very similar to Priam (although he actually seems to have been on the Greek side). And we somewhat confidently know that Troy had a king named Alexander (Alakasandu) which happens to be the Greek name of Paris (of course there were probably many Trojan kings carrying that name over the years).

Of course that doesn't mean much, the Iliad is a mishmash of different historical events from different periods (based on the descriptions of weapons, armour, political systems, cities etc.). There is probably a massive accuracy gap between oral history like the Iliad and written one (including the Hebrew bible).

The problem is, we don't just have an absence of evidence, we have evidence of absence. The area has been widely excavated, and there is a clear continuity of settlement with the same pottery, culture, and religion. There is simply no trace of any large-scale population movement. As far as we can tell, the same people continued living in the area in the same way, worshipping the same gods (still plural for way longer) with the only large change being the yoke of the nearby great powers going away with the collapse.

This of course doesn't mean that there cannot be a trace of truth in the story! It just has to have been morphed substantially over time. For example, it was common in the time to kidnap and move foreign nobility and artisans, while no-one much cared about the identity of the average farmer or goatsherd. It could well be that "the people of Israel" who were kidnapped meant the people who actually mattered, ie, a fairly small upper class group, who could move from the Nile valley to the levant without leaving much trace in either society.

I'm still personally partial to the observation that the story seems to originate during the Babylonian captivity, and the situation of the story greatly mirrors the conditions they were living under, but while complaining about their Babylonian overlords was probably not allowed, writing stories about the plucky underdogs outwitting the horrible Egyptian overlords with divine assistance was fine, even if it contained themes of returning home and of liberation from foreign rule. (Note that Egypt was the main rival of Babylon in this period, and the Kingdom of Judah was on-again off-again vassal of the Egyptians. The captivity was party imposed to prevent this relationship from continuing.)