The Dogeship is limited to members of the Venetian Patriciate, the oligarchy of noble families listed in the Libro d'Oro. By the time of Lorenzo il Magnifico those families could trace their lineage back several centuries, the Medici couldn't even do that in Florence let alone of foreign city like Venice.
So you would need a POD far enough back to change the government of the Serenissima such that foreigners were elected rulers a la the old tradition of the Podesta' (never used in OTL Venice), but that would likely butterfly away Lorenzo or the Medici's rise to power. The other way would be to have Venice be conquered somehow by someone who would, for some reason, want to put Lorenzo in charge. I can't really think up such a scenario, at least not a particularly realistic one.
The bigger problem is that I doubt Lorenzo would even want to be Doge. In Florence he was de facto absolute ruler of the city, even though he rarely held elected office officially. But the powers of the Doge are extremely limited, checked by the Great Council among others. Venice was, after all, an oligarchic Republic not a dictatorship or monarchy. And perhaps most damning is that Venice had strict policies against developing a cult of personality or personal aggrandizement. Not just sumptuary laws but also things like forbidding statues of rulers in public places. The dignity of the Doge was tied to the office not the person holding the office in away that was more Byzantine or Late Roman than it was monarchical, in the Western European sense.
Though I suppose if we went with the second scenario and Venice was conquered Lorenzo could be made into a and absolute ruler type of Doge but again, I can't find a realistic way of making that happen.