Just saw posts about video I linked. The video is about fragging and demonstrates one way to "assist" the process of fission/pedal laceration by cutting mushroom in half while it is in the tank which is meant to multiply the number of established mushrooms as I do it. Sometimes it is easier to cut them into quarters while they are attached to rockwork.
To remove them from a particular area, I have found it necessary like video shows to chip off part of rock underneath the mushroom and have also had some luck removing part of mushroom using plastic tweezers and now will try edge of credit card as vetteguy suggests. I've also read that any strategy which makes the mushroom want to release its grip and move (e.g. kalk paste, increasing flow, increasing light) has been useful at times.
From my experience I think the type of rock the mushroom is on plays a role in how easy it is to remove and, of course, since mushrooms can withstand rough treatment I don't expect to remove it without cutting, tearing, etc., some of its flesh, all with the expectation that it will regenerate quickly.
Good luck and a good reason not to place mushrooms on rock which can't be removed from the tank, lol.