My two cents: perhaps the Ottoman Raj starts as a way to outflank the Persians by co-opting local Muslim rulers and gaining influence by the back door, i.e. providing the military expertise for them to triumph over their neighbours?
In 1555 the Ottomans won a war against the Persians, providing them with the southern coast of the Persian Gulf, which would open up the area for development by the Ottomans. Perhaps, after Lepanto, a more chastened Ottomans might decide instead to focus on building up their Red Sea and Persian Gulf navies instead of challenging the Europeans on land. This trend might also be intensified as a result of further catastrophic collapse placing Istanbul on the front lines and necessitating a moving of the capital to Baghdad or some such.
The Ottomans, strengthened by their Indian acquisitions, might be able to secure a direct land route to India in later Persian wars, permanently changing the balance of power in the east, and continue their conquest of India. In this case we would have an empire centred on the Persian Gulf with a weird Egypt-shaped tumour to the west and a weird Anatolia-shaped tumour to the north. I dunno, it wouldn't really be the Ottomans if they didn't have Anatolia, would it?