A neat trick for Berghia - leave them in the shipping container for a day or two with a little air inside. Flush the container a couple times with tank water to reduce wastes. If you allow them to lay eggs in the container, you now have 2 generations of Berghia to go after the problem, just in case something eats the adults you bought. Peppermints will eat Berghia, so don't try both. I'm also suspicious that 6 line wrasses are eating them as well - why not, since they eat zoa and monti nudis. Also, it takes a couple generations of Berghia to eliminate Aiptasia. My guess is the babies are better at sneaking around without predators getting them. Berghia are hermaphrodites but cannot reproduce alone. Technically, you only need 2 to get started, provided they can lay eggs before being eaten.
I added 20 to 5 different tanks a few months back. I'm just starting to see an effect on the Aiptasia population, but they are starting to disappear quicker and quicker. The most successful was the last tank where I just added the container with egg masses in it to a refugium that had a few Aiptasia. Not only did the babies eat all the Aiptasia in the refugium, they spilled into the display and ate all the Aiptasia there.