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.