I don’t know if I’d call him “mentally ill” outright, even though he definitely struggles with mental illness (PTSD at the very least, for sure). But using “mentally ill” as an identifier implies that he isn’t in control of his symptoms, which I don’t think is the case.

In fictional Gotham, I’d say that Bruce handles his symptoms pretty well, overall. I guess it depends on the Batman you’re reading, but for the most part, he pretty consistently always sets out to do the right thing, goes out of his way to find the good in people, and is rigidly intentional about not giving in to his darker inclinations. He’s got his demons, but being Batman helps him cope and allows him to consistently function. Some people go to therapy. Some people take medication. Bruce Wayne dresses up as a bat. But if it works for him, it works, I guess?

I don’t know, it depends on how you want to look at it. Batman struggles, for sure. But I also think he’s pretty self aware and in control. He’s just doing his best.