Countersue? Not really, unless you're still owed fees or something. But suing to avoid paying the bill is more common with the people who sue my legal-malpractice clients, as opposed to the medical-malpractice ones.
Typically, attorney fees are the responsibility of each party, but many jurisdictions, including mine, have various methods for shifting them from the losing party to the winning one, such as when the case is frivolous, or if one side does worse at trial than a pretrial mediation case said the case was worth (i.e., if the plaintiff could have offered to settle, didn't, and then loses entirely). For the most part, though, your malpractice carriers pay your attorney fees (if your policy says otherwise, which would surprise me, you'll want to consider a different one, I think). I believe some policies also cover stuff like your lost time for depositions, hearings, etc., but that's not really my wheelhouse.
But suing for the indignity of being sued is not a realistic option. The closest causes of action are "abuse of process" or "malicious prosecution," but generally speaking, you're more likely to prove Bigfoot exists than either of those causes of action.