But control over Bithynia would be critical for any such polity, no?
I don't see how the Ottomans manage this just by controlling Bithynia and without their Balkan possessions.
Neither are essential for either. It just isn't easy to do without. But peace and neutrality can be enough.
Full control over the straits is great for either power. But if the Romans are strong enough to prevent the Ottomans (or anyone else for that matter) from coming over, then great.
Both powers can benefit from peace between them (although losing the Ghazi is a loss) - so you could have initially the Ottomans go a-pirating. Lots of ghazi ambushing trade ships, looting and running to shore. Sates the Ghazi a bit, makes the Ottomans quite rich. This can provoke either a war (lets assume a stalemate), but potentially a small Roman victory that forces a peace and trade treaty - the Ottomans get a preferred trade status, which includes lower tariffs in the Straits/Dardanelles (same for the Romans), in exchange for long term peace. That benefits both, and can enable both to dominate their local rivals more easily. Romans hire Ottoman troops as Mercs, Ottomans get Roman assistance in Anatolia, etc.
The Romans think they're in control, but eventually lose it because ... hello? Ottomans. They become more equal partners and the Ottomans aren't a rogue pro-Roman entity in Anatolia, but the leading Turkish power, with a strong economic backbone via trade, the same backbone as the Romans. There will be factions that disrupt this relationship, but for the same of the TL, they're minor disruptions at best.
So both have that free hand in a secure border that allows them to expand more easily - eventually both dominating their respective regions.
The Ottomans can then continue from there. More mercantile and Pro-Roman than most Muslim Empires, and still secure in that western border for now.
Eventually, that will break down, but if the Ottomans have refocused in the middle east, even a devestating Roman victory will only force them to cede lands on the western border - they still have trade in the Levant, control over the Spice Trade until Europe goes around Africa.
TL;DR mutual trade, peace and secure borders can jump-start both realms immensely. It'll fall apart eventually, but that gives the Ottomans plenty of time to dominate the Middle east