I had them several years back, but luckily they were mostly on my rock work so they were easy to suck off with a siphon hose. The thing shown on that melevsreef article is what I ended up doing so I could just dump the siphoned water back into my tank.
Seems to me (and just thinking out loud) that if you just run a "gentle" siphon you'd be able to suck the buggers off the mushroom without harming the mushrooms too much. And even if the mushroom retracts on you, then it seems like the flatworms would be ejected off the 'shroom since they're not going to shrink in size like the mushroom would.
I remember when I was dealing with it, they seemed like they were never going to go away. But eventually I got the population down to a certain level that they disappeared within a few days. But it took about a month of constant battle to get to that point...