patience and persistance.
personally i'd take the time to slowly remove them completely, or to a controllable population.
use razor, scalpel, coral cutters to remove by chipping a bit underneath base removing with bit of rock underneath. you need to make sure to get the base out or it will just regrow.
small ones, you can cover with epoxy as well.
i am not aware of any natural predators.
i've prepared myself to do the same. i made an informed decision to keep mushrooms in my reef and i'm prepared to have to deal with their spread.
i keep an eye on growth/spread and especially for any that have detached and found their way to somewhere i didnt intentionally place them.