Suppose I have a tennis racket spinning on its second axis of rotation such that it is subject to periodically flipping over on its spinning axis, as per the Dzhanibekov effect (explained in animations here).
Now, make this tennis racket be nearly planet sized -- has to be "nearly" because a planet has enough mass to pull itself into a sphere, and a tennis racket is not a sphere. If it bothers you to picture a tennis racket, consider an actual astronomical body: 486958 Arrokoth, a Kupier Belt object. Arrokoth doesn't spin on its second axis, but given a good enough pounding by an asteroid impact, it could spin on that axis. This planetesimal is solid, not liquid, so it does not dissipate heat, so it's rotation about its second axis is stable.
So this planetesimal isn't big enough to become a sphere, but it is big enough to have a reasonable gravity field, so another smaller body is capable of orbiting it: a moon. The moon orbits the barycenter of the object, which, for the purposes of this discussion, is right at the neck of the handle.
Now, here's what I want to know: When the tennis racket flips over, what happens to its moon? I cannot figure out whether it just keeps orbiting without noticing (because it sees the racket as a point mass for the purposes of orbit) or it get dragged along with the racket or it flies off into space. Or maybe some other effect.