OK, so we've got the start of our rough TL here -- sometime after the marriage of Theodore Laskaris to the daughter of Alexios III in 1199 (so 1200-02), the Emperor trips and falls down the stairs, cracking his skull and dying. After a brief power struggle, the husband of the late emperor's second daughter emerges victorious, and is (presumably) crowned Theodore I.
The Fourth Crusade progresses as OTL, and the army of Venetiains and Crusaders arrives at Constantinople in 1203 looking to put Issac II back on the throne. The difference is TTL, the defending emperor does not flee the city; as a result, the nobles of the city don't open the gates or agree to put Issac back on the throne, and the city continues to defend itself. The Varangian Guard does its thing, and the main crusader army is held at bay, even if some Venetians manage to take a tower or two and/or get over the walls.
How TTL's Fourth Crusade comes to an end exactly I'm not sure of, but it does seem that the Byzantines are capable of keeping them out of their city (if for the most part). I imagine that relations with much of the Catholic West (the pope, Venice, the HRE, etc) will be very chilling following this confrontation, which will not make things easier for our Theodore; adding to that the border issues with the Turks and the Bulgarians and rebuilding the navy, the ERE has some serious challenges ahead.
But on the plus side, they're in a much better position to meet them.