Out through the bottom...it worked!!
I knew coming and asking here was the best thing I could do.
I had originally played with one of the clasps on the bottom and got one out, but it seemed like it didn't do anything - it looked like the bottom was one complete formed piece of metal. Got both clasps loose and pushed the mirror out the bottom - there was a black plate under the mirror which made it look like it was all one piece with the base.
Ah, wonderful. I am so very, very happy!
Thank you, thank you, thank you.