Hi,
I just figured out how to remove the base from the holder on the window. There is an oval spring that holds the base in with two prongs at the top behind the metal ring. I used an allen wrench to move one side in while I slid the base out of the holder half way, then used it on the other side to get it the rest of the way off. A pair of needle nose pliers may work. Note, the screw that fits at the base of the mirror is very tight, but once I got it loosened, I was able to re-attach the mirror to the base, and the whole thing just slid back into place.