Is there any evidence that the Sea Peoples or other invaders had Iron weapons, and that this was the reason they could successfully conquer the established civilizations of the Late Bronze Age?

No. For one, the Late Bronze Age civilizations were already utilizing iron, though bronze was preferred. I touched on this with respect to the Hittite empire in To what extent did the Hittites make use of iron?

More to the point, iron weapons were not really any better than bronze weapons (indeed, it's arguably the other way around). The main advantage of switching to iron is that iron production was not dependent on long distance trade routes, whereas bronze production required importing tin from Cornwall and Afghanistan. There's always more to be said on this, but u/wotan_weevil wrote about this in If iron is really better than bronze in every way, why did it take so long for us to move on from bronze? and How much really better was Iron than bronze?