Could the Romans punch through to Mesopotamia again this war, Time of Troubles style? If Ibrahim's army gets cut off and broken, the Ottomans won't have enough time to muster up new field armies at the front to stop border incursions. I don't know how their logistics arms will look with the Anizzah so weakened, but Napoleon-style local forage is an option.
I think their objective is primarily a quick end to the hostilities in the East. A drawn out campaign into Mesopotamia is probably out of the question while a good chunk Latin Europe marches on Constantinople and the burden to the Roman exchequer is at an all time high. If or when Ibrahim is decisively defeated I assume the Romans will try to sue for peace to transfer troops back West. Imo, at the most they might make a half-hearted attempt in the general direction of Mosul to force Ibrahim to the table and get the best possible terms.