One of the reasons mosquito populations are out of control is that we've killed off a lot of their predators. No mosquito anywhere is a keystone species, and you would only need to wipe out the vector species. Other, less harmful species of mosquito would fill in nicely with less competition.
At least, that's the theory. Previous theories included introducing mosquitofish to eat the larvae, but that backfired because the moquitofish are aggressive and don't eat as many mosquitos as local predators driven off by the mosquitofish.
Bats eat tbem, but they won't go extinct without mosquitos as food. There also aren't enough bats currently to keep the mosquito populations under control due to other factors preventing abat population boom.
I gotcha, I thought you were saying that bats would suffer if the mosquitoes were eliminated.
Yes, anything we do to improve bat populations would be good for a variety of reasons. But I doubt that any one bat species would make a dent in the specific mosquitoes that act as vectors for disease. Plus, if you start introducing bat species to non-native habitats, you run the risk of repeating the mosquitofish catastrophe.