There is also much evidence for a westward artistic influence from the Iberian Peninsula impacting Sicily during the same time:
And things like checkboard patterns, chevrons, wavy lines, are all attested in earlier Neolithic rock cut tombs in Sardinia.
By the way, I'm not denying any possible connection with the Eastern Mediterranean or the Near East in general. The Cetina culture for example connected the Aegean with South Italy, Sicily included. Asian elephant ivory was exported westward and it arrived as far as Iberia and Sardinia during the third millennium BC. An Eastern Mediterranean type axe made with Aegean copper was found in a Rinaldone burial in Latium. And how not to mention the famous ossi a globuli? They were found both in the Eastern and Western Mediterranean, including the much discussed Troy itself! too bad that they date to roughly 2200 BC, not 1200 BC. So there were more or less direct contacts between the two sides of the Mediterranean already by at least by the third millennium BC, though it's hard to say how exactly they occurred and who the main agents were. DNA seems to support that there were Westward movements which reached Italy and beyond during the third millennium BC. But for Sicily for example, there is also some DNA evidence that seems to support migrations from the Iberian Peninsula too. So the opposite is also true.
The problem is that when we discuss the Trojan migration we're talking about another period, the end of the Late Bronze Age, not the beginning of the Bronze Age or the Copper Age, and about a specific migration from Troy taking place. There isn't any material evidence for that. There is clear evidence for some Eastern Mediterranean groups traveling westward during the Late Bronze Age, especially Greeks and Cypriots, but not Trojans, and seeminly mostly to trade rather than to settle.