Have had zero luck with peppermints and berghia. What I have done in the past that's worked is use kalk - but not the kalk water, the pasty goo that forms at the bottom of my mix by putting in too much kalk powder (to the point that more kalk doesn't dissolve). I use a very stout syringe needle (had to get it at UFA - it's for livestock or something - the hole is easily 1mm diameter - anything smaller just jams) and then inject the paste into the aiptasia.
Haven't tried vinegar, that sounds easier. I'll have to try it.
My CBB eats aiptasia like there's no tomorrow. Unfortunately it's only a solution for a tank that can fit a CBB and you have to luck out with a CBB that will eat aiptasia (and not starve on you because they are fussy eaters). And unfortunately it won't work for a small tank because you need a decent size for them to swim around. And the other downside is that they are not always reef safe. My CBB eats plenty of pricey LPS pieces (blastomussa, acanthastrea, dendrophyllia have all had to be taken out of my tank so far
).
If it's a rock you can move, you're welcome to put it in my tank for a couple of days .. I'll guarantee mine'll clean it up within a day or two and then you can have the rock back.