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.

"Sheltered from the reality of inter-organization politics" is a pretty wide bucket. I've been on teams that constantly fight with each other to get resources and executive attention, and I've been on teams that collaborate with each other to make the best decisions taking into account everyone's constraints and capabilities. In my experience there's a huge qualitative difference that doesn't have much to do with hiding or disguising anything.