Sea Peoples weren't really a "mysterious people", as much as Late Antiquity Barbarians weren't one entity. In all likeness, they were composite, possibly rival groups.
While a fair deal may have come from Anatolia directly, there's little reason to doubt that some Greek (Myceneans or peripherical) and other Balkanic peoples (Proto-Phrygians?) participated to the general movement (Aqawasha/Ekwesh/Denyen, etc.)
It's even quite possible that peoples from Adriatic, Ionian and even Thyrrenic coasts not only joined the bandwagon but pushed it on the rails, as hinted by some names (Shardana/Sherden, Sekelesh/Siculians) or by recent discoveries on Bronze Age Central Europe which hints to great local disorders due to climatic changes (among other reason).
The "long migration" from Central and Northern Europe let me somewhat cautious, tough : there were "Sea Peoples" regionally present before Ramses III (during Ramses II's reign for instance, as he had a "Sea People" personal guard), and inner European migrations can't explain all movements : Mycenaean decline, for instance, isn't that well tied to these with Dorian Invasion being more of a convenient and systematical explanation to fill gaps but not well proven. Social upvehal on an already decling Late Bronze Age, tough... (Altough these coups and revolts could have found some support among foreign or local constitutions of SP : again, it's really possible that some of these peoples were issued from the old Mycenean sphere).
Some were known as "settled" peoples, as Lycians (Lukkas) and the identification with places (Shakalusha with Sicily) doesn't sounds that definitive to me : it could be as well tied with other places (Skakalusha with Salawassa/Sagalassos in western Anatolia)
Eventually, I'm more convinced by a mixed Adriatic/Aegean/Anatolian population with central European elements, at least for what matter to the bulk of Sea Peoples; a bit like the "long migration" model for Late Antiquity Barbarians is dubious, and should be considered as a long ethnogenesis on the trip.
So, what are the effects of an avoided or diminished Bronze Age Collapse?
The Late Bronze Age Collapse have probably many other causes : climatic change, greater interdependency, warring polities (especially for Hittites and Cyprus, or Egyptian and Syria), whom Sea Peoples are but another symptoms. The destructions attributed to them, furthermore, shouldn't be exagerated : either due to local powers, or local peoples (such as Kaskas in Anatolia).
The set of climatic changes, particularily important inter-dependency in a declining period with renwed conflicts would probably let his mark even if climatic changes wouldn't have led to inner migrations from Central and Southern Europe, possibly feeding these troubles, and spiraling out of there.
A lesser Sea Peoples migrations in Eastern Mediterranean basin isn't unthinkable,
but as much as it had been discussed, would require an early PoD (I myself tought about a Bronze Age civilization in IOTL Tunisia, as a possible derivative and structuring element in central Mediterranean basin, as hard of a change it is) that wouldn't get rid of, not only the decline of Late Bronze, but migrations as a whole.
