Where do you think values and cultures come from, if they aren't set and reinforced by managers? Yes, there are degrees to this - a frontline manager at FAANG has a lot less power to shape than a VP at a 100 person company. But you have the power to shape, to some extent. And you are the person people on your team will look to for cues.

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.