To keep a hold of this any Italian Empire would need to hold the neck. Genoa has a strong fleet, but to hold more than a few coastal enclaves, they need to prevent the hordes from assailing the Crimean Peninsula by land. This requires guarding that lagoon-filled neck. Eww.
Perhaps if they decide that it would be worth holding, they could use the natives, and any immigrant Italians as future manpower. But they need resources, something to make defending the neck worthwhile.
Any significant natural resources on the peninsula that would entice Genoa to stay? I can imagine they may exploit the land for farming if that is within their means, but you're looking at a merchant republic becoming very territorial, which may end up leading to a duke or count or titular leader.
Though that would make sense if they genuinely sought to unite Italy.
Problem is for Genoa is they would rely on good relations with whoever controls the dardanelles. Which if they are defending against and fighting the tartars, means they become very tightly involved with the Byzantines.
In addition, the Byzantines would be very unhappy with someone empire building in their backyard. Especially someone who is rightfully a Roman vassal.
Perhaps we have a Genoa that becomes a co-operative "Capetanate of Italia and Taurica (Crimea)". I'd love to see an Italian Empire that essentially usurped the Greco-Roman byzantines and made it Italo-Greco-Roman Empire, but I'd more likely expect a Byzantine-Endorsed Italian power growing and eventually asserting their independence.
But without access to those straits, the Italians (Regardless of who) are going to find it impossible.