Could Genoa get sucked into the war earlier ITTL because they are seen to be in a weaker position than OTL?
Genoa's entry into the war was the result of concessions given to Sardinia in the Treaty of Worms in order to bind Sardinia into a formal alliance with Austria and Britain. The problem was that one of the territories which Austria conceded was the Maquisate of Finale, which was not theirs to give - it was Genoese.
Finale had been granted to Genoa in the 1713 Treaty of Utrecht. Maria Theresa seems to have held that it had merely been
mortgaged to Genoa and could be redeemed at her will, and the Treaty of Worms included a provision by which Charles Emmanuel would receive 300,000 pounds from the British to "redeem" Finale. But they had never asked Genoa whether it
wanted to part with Finale for 300,000 pounds, and Genoa insisted (correctly, it would seem) that they had
purchased, not
mortgaged Finale and that the Pragmatic Allies (Prallies?) had no legal basis to demand it from them. The irony of it all was that Maria Theresa, who had been absolutely enraged that anyone would dare try to steal a piece of her patrimony, had now blithely consented to Genoa being carved up on the basis of naked avarice. Even some British statesmen admitted the dubious nature of the Finale clause when they read the treaty (William Pitt described it as "such an act of injustice toward Genoa as must alarm all Europe").
Within weeks of the Worms treaty, word of the planned "redemption" of Finale leaked out to Genoa. They had remained neutral thus far, knowing very well that they were a weak power and would be best served by staying out of the fight. The intended cession of Finale, however, meant that they now had a vested interest in a Bourbon victory, for that was the only way by which they could avoid losing territory. It logically followed that if they were to be forced into the Bourbon camp they might as well get something out of it, and thus their diplomats been negotiating Genoa's entry into the war in exchange for the cession of some choice pieces of Piedmont.
The main reason it took them until May of 1745 to sign an alliance with the Bourbon powers was because of the credulity of France. The French spent much of the WotAS hoping against hope that they might be able to get Sardinia to switch sides, which made them easy suckers for Sardinia's brilliant diplomatic game. Obviously signing an alliance with Genoa which offered them Sardinian territory was incompatible with flipping Sardinia to Team Bourbon, so France held off on any formal arrangement with Genoa until the Spring of 1745 when the French and Spanish armies converged on Genoese territory and it no longer made sense to keep waiting for Turin to change its mind.
I don't think Genoa's position on Corsica (which was not strong IOTL either) will have much of an effect on this. Certainly a weaker Genoa won't prevent the Pragmatic Allies from signing away Finale at Worms, and once that happens it's only logical that Genoa will start gravitating towards open alliance with the Bourbons. How soon they get there depends less on Corsica than what's going on at Versailles and on the battlefields of Italy.
It seems the last citadel left for Theodore to possibly take on his own would be Bastia, the rest would have to wait for the British Navy.
Bastia is going to be a problem so long as the Genoese still have 1,500+ regulars there (and perhaps more now that the Ajaccio garrison has been evacuated). That's not a force the rebels can easily match. That said, however, after the treaty of Worms the likelihood of war on the continent caused the Genoese to withdraw men from Corsica at a precipitous rate, well before they actually joined the Bourbon alliance. These are OTL figures for the number of Genoese regular troops in Corsica:
November 1743: 2,523
January 1745: 1,375
September 1745: 1,078
May 1746: 760
May 1748: 715
Such drastic troop cuts would be much more dangerous ITTL given the greater strength of Theodore and the nationals, but the Genoese don't have much of a choice. They clearly viewed the loss of Finale as a graver threat than the loss of Corsica. Corsica, after all, could always be reclaimed after the war with foreign help, but if Austria won the continental war and Sardinia took Finale the Genoese would be powerless to reclaim it. The question
then becomes whether they try to hold on to Bastia, Calvi, and Bonifacio with such meager forces or whether they abandon one of their posts - presumably Bastia, but the capital's loss would be a hard blow - to try and further consolidate their remaining forces and keep their "toehold" on the island.