I heard of a method but have never tested it, so best of luck if you try it......
Place a thick towel folded once over a wooden work bench.
Take mirror housing in hand, glass side down.
SLAM it into the towel / work bench. Repeat as needed & check to see if it is loose enough to pull out in between. The dowel pin will eventually release from the housing.
The guy that told me about it said he has not broken the mirror in one. Yet..........