Odds are, considering Lajos' character, that the Hungarians will leave the Lascarids to fend for themselves.
And it will be only fair - after all Theodore made peace with Louis of Taranto and let the Hungarians alone to deal with the mess. A very expensive mess.
Having said that, I doubt that Alexandros will see it that way - he has a bigger ego than his father and was also recently defeated by the Venetians. At some point. I expect to see a rapprochement or detente between Syracuse and Naples. I doubt the Hungarians will get any help in Italy in the future. I also expect the basileus to deal directly with the Camponeschi and not through their official overlord. After all, Alexandros has a lot of daughters to arrange marriages and Abruzzo will be a key partner.
Moreover, Abruzzo has a good reason to be relatively hostile to Venice: the Venetians were doing their best to ensure that Venice was the entrepôt of the Adriatic. Therefore, they were trying to dissuade ships from trading directly with the eastern coast of Italy, Abruzzo included.
Since this war had not been in defense of an existential threat, Alexandros II will likely find it hard - politically - to raise the subvertio generalis, very hard to convince the parliament to consent to it without a significant backlash. Take new loans from Genoese banks is less risky, politically speaking, so more likely I think.
Was privateering something yet in this era? If Alexandros cannot raise too many ships without raising taxes sky-high or huge loans, he could encourage/incentivize private actors to prey Venetian shipping on his behalf and pay themselves in loot.
The other solution is internal loans from the burgher elite. The peasants wouldn't care about a war with Venice. But the burghers will have the opportunity to become the dominant mercantile power in the Mediterranean. It is a stakeholder group with a lot to gain by digging deep in their pockets. In any case, they can enforce the creation of a bank to protect their rights and ensure that the Basileus pays them back.
Privateers were a thing and Michael Palaiologos used pirates/corsairs in his policy and so did the Genoese and the Catalans. By that time, most of the venetian trade was carried by cogs or in general round ships. The galleys would carry spices and silk, but most of the merchandise needed more spacious holds. Geography dictates that sicilian corsair fustas will make a killing.
In the 14th century they were also letters of marque. To quote "Piracy and the Venetian State- The Dilemma of Maritime Defence in the 14th century":
The letter of marque was similar to patents granted to corsairs, but it was issued to a specific individual in response to a particular incident and valid only for a limited time. The reprisal could allow him to assault either the particular vessel which had originally attacked him or all vessels originating from the aggressor's territory
Alexandros can very cheaply acquire dozens of Genoese