I have two giant wrasses which have been with me 5 years (melanurus and leopard) and they don't bother Berghia. I'm sure some fish may be problematic, but I've had berghia in my system for 2+ years and they live despite having almost zero aptasia anywhere and hungry wrasses. I've introduced Berghia twice. Once on my 250gallon I did about 25 medium to large, they took them all out over 3 months and then rebounded. Second time I did like 10 small to medium, and they took longer, like 4-5 months to take care of everything, but it's been over a year or more since that time and I feel like it's a balanced level which keeps them in check.
Just make sure the lights are out and you give the berghia a head start. Put the cup of berghia somewhere very low flow and let them escape. Probably give an hour or so in calm water. They MUST crawl out on their own accord or they're goners. You won't see them again except once in a while. It's been a very, very long time and I saw a giant one on the glass the other day.
In the past, I've had the best luck using peppermint shrimp and aiptasia eating filefish. I keep both in my systems, all the time, just in case. I've had no luck at all with any of the products out there. They always came back worse than before. The most important thing to consider when wiping out aiptasia is to hit it in the spots where it hides, like your overflows and sumps. One trick is to divert your overflows and fill with a vinegar / water solution, letting it soak over night, then drain it all out. Or, introduce nudi's that eat the stuff, into these areas.