No, if the board doesn't believe it can continue to serve the company and meet its fiduciary duty, its obligation is to resign. CEO is an operational role; the board by design is not. It's a very big deal to "fire" the CEO, and doing so when you don't actually have the voting authority to follow through seems pretty close to malfeasance.
I mean it sounds like “board which does not actually have the legal ability to fire the CEO” is a fundamentally defective concept and shouldn’t be allowed to exist in the first place. But once it does and you are in that situation, I think you are obligated to make the best attempt you can at your nominal duties. I have no idea where you’re getting “malfeasance” from at this attempt.