Just about any piece of a toadstool will regrow into a full toadstool if the conditions are right. I have found the frags to grow rather slow though.
Depending on how you frag it, what's left in your wife's tank could look pretty ugly.
If you want to maximize the number of frags and still keep the basic shape of your wife's, you could cut a 1/2 to 1" ring all the way around the top. Then, you take the O shaped ring you made and cut it into 1/2 to 1" sections. Eventually, each section will grow into a complete toadstool. Your wife's toadstool will be mad for awhile; but, the disk that's left will expand outward again.
If you don't want to mess with your wife's much, and you have all the time in the word to grow your frag, you could cut a little bit off the foot on the top spot (say about a 1/2" piece). After awhile, you'll have another one, polyps and all.
As was suggested, frag it outside the tank as it will slime up a lot when you cut it.
Regarding the mushroom, the best way to move it is to dig out the bottom of the mushroom from the rock with a knife or flathead screwdriver. By taking a thin layer of rock under the mushroom, you'll remove it completely; and, you'll have something to attach the superglue gel to when you glue it somewhere else. If you cut the mushroom or tear it in any way trying to remove it, if the conditions are right, each piece will grow into another mushroom. It sounds like that is not what you want.
Best of luck,
Roy