If its a baby mushroom.... I would probably not want to cut the stalk off, because it may not reattach and just melt away... So I would chisel it out of the rock and glue somewhere where I want it.
It would be easier for you to move the big mushroom (just cut the stalk). Once you cut it off, the foot will grow into a new mushroom. The top, put it in a plastic cup, with some live rock rubble in it. Put it in a place where there is low flow, and cover it with some veil mesh, so the water doesn't get stagnant (bridal veil). Wait about a week or so. After a week, it should be attached to a piece of rock. Take that rock, and glue it where you want to.
If you want to frag the mushroom, cut it in half through the mouth. Dip in lugols to prevent infection, and repeat the stick it on some rock rubble. Make sure both pieces have some of the mouth. I'd start cutting from the mouth, and move outwards, to be safe. Try to do all of this quickly. The faster you go, the more likely you won't kill your shroom. Also clean clean clean everything, and get a sharp sharp sharp knife.