How to replace a mirror in the sunvisor without destroying it?

If its the cloth one. You will never get it apart without destorying it.

If its a convertible vynal one, then yes, you can pop it apart on one side and slip the mirror in.

The mirror frame has plastic pins in the back of the frame. There is another piece of plastic or hardboard on the under side of the fabric. The frame is secured to the backing by some of the plastic pins from the mirror frame being melted and folded over the backing.

In short an auto upholstery shop will have to tear the visors apart and then sew them up so that you can swap the mirror.