BÜ wants a declaration of assignment. How to avoid it?

Yes, but only if the general contractor continues to insist on a guarantee/assignment (?). I would not just barge in with the goal of a conflict-free solution.

Not barging in, but the performance bond is owed to the client and has nothing to do with the assignment. It is his full right to demand this bond.

I would aim for a "reset conversation." The focus should be forward - not on who did what or how. A handshake and move on. If it works, you already have the experience of how to deal when things get stuck. Apparently, there is contact with the owner of the [GU].