"The company" is nothing, except what leadership makes it. As a manager, that includes you. If everybody chooses not to care about org problems? Guess what, that's a manifestation of sociopathic behavior.
So, act accordingly.
So, act accordingly.
Mental health issues are for life. It's not worth the risk.
Which means it's a question how big the gap is. If you can cover it with team-local culture, by all means, straddle it. But trust me, if you completely abandon your values, your mental health won't do well either.
I'd suggest that a severe/unfixable values mismatch should trigger a job search on your part, because acting out values that contradict yours rarely ends well for your health.