If Stannis had managed to take the city, wouldn’t the Lannisters still have a claim to the throne in Tommen? Or would that not be enough to keep the alliance with the Tyrells together when there are still other rebels?

1. That depends on whether they go looking for Tommen and manage to find him. 

2. Even then, a lot of lords would think twice about supporting a boy king who’s just lost the capitol against an adult male who’s just won it.