How much of your navy would you send to fight someone else’s battles? 

Even with the alliance, none of the Three Daughters had been attacked directly by Rhaenyra (although they had beef with Daemon Targaryen, certainly), nor had direct interests at stake. It was really more of an alliance of convenience, a hope that victory would bring with it rewards, advantages, a renegotiation of the Stepstones perhaps. 

The Battle of the Gullet is rather weird on the dragon angle. Given what happened in the Battle off Gulltown, you’d think the battle would be one-sided in the extreme. My guess is that GRRM wanted the battle to go one way but had already decided the numbers and dispositions of dragons.