This depends on whether Serge's body was found and buried on Cape Howl or not. If there was a corpse, then the dimensions split during or after his saving. If there was no evidence, then the dimensions split the moment Kid went back in time, and somehow Serge simply wasn't part of Another World.
I doubt we'll ever know whether he truly
existed in Another World or not, unless anybody with a decent Japanese translation can somehow dig something up about a corpse (no pun intended). Either 1) Serge's body was actually located and buried on Cape Howl, or 2) the townspeople in their engraving on his tombstone ("What comes from the sea / must return to the sea") hinted that his body must have drifted away with the tide.
I would think his complete non-existance in Another World would pose a problem, though, since how could FATE possibly figure there would be a living Serge crossing the dimensions if there was no tangible evidence that Serge even survived the encounter with Lynx in the other dimension? Unless Another FATE can view Home World's Records (which wouldn't be completely out of the question, since we know FATE can see into La Trésor Interdit, for example), there would be no assurance that Serge wasn't just tossed into the DBT as a casualty of the rift, never mind his status as the Arbiter.
Then again, perhaps FATE (through Lynx and his influence on the Acacia Dragoons) planned for the impossible as a "just-in-case" measure, sending Karsh, Solt, and Peppor on what seemed like a wild goose-chase for the ghost of a boy whose body may or may not have been buried on Cape Howl. Or, the most unlikely of alternatives, FATE could be a cross-dimensional program with complete awareness, which doesn't even begin to explain why its power is so limited in the Home World.
Also, whether Lynx or FATE made the final decision to destroy the Dead Sea is, basically, irrelevant. They're of the same mind anyway. For all we know, it could have been an automatic defense mechanism triggered upon either Miguel's death or as a precautionary step in case the Dead Sea's Frozen Flame was in danger of being discovered. In either case, it triggered the Angelus Errare in the Sea of Eden allowing Serge to travel to Chronopolis, which FATE, had it been active in Home World, would have certainly prevented.