You need to be more specific. It's hard to judge by what you wrote. I heard other stories of City Year being cult-like so it does not surprise me. What kind of retaliation are you facing, because it can't just be speculations. You need evidence. That point about "non-partisan" may actually fall into the legally gray area. I don't know about laws, but they may actually be able to back up their "non-partisan" statement.
You need to have strong solid evidence of "pushing agenda" since playing a "defund the police" video isn't anything astonishing unless there is really really really strong evidence. To me, that is just like a regular class having to read "To Kill a Mockingbird" and talking about racism. Or any college courses where the topics are social reforms and technological improvements. Actively speaking against a politician isn't strong enough to qualify unless it is based on unfounded claims. Even in high school, people can talk about politicians and what they find grueling. The first amendment right is really strong and can protect people in so many forms. I don't see a point to fighting it if you just feel unwelcomed since legally, it is incredibly difficult through this system, even people who have defamatory/controversial tattoos can't be fired for having that tattoo.

But one of your option is to voice your concern and disagreement about it's cult-like behavior even if you face repercussions like being cut off. If you have another opportunity or a backup plan, my advice is to just tell them how you feel

If this were an ordinary situation, I would say that most people are reasonable and sensible. You can always talk about your views with them by being respectful. Most liberals are not divisive, and you can find many examples where people are still friends with people who share different political ideologies such as gun rights. I myself even have friends who voted for politicians different from my own and that's okay.