Berghia is a general hobbyist term which refer to a group of nudibranchs which feeds primary on various types of anemones (and anemones only). If you can obtain one (either locally or from an online vendor), it will eliminate your pest anemones. When all the Aiptasia are gone, the Berghia will starve so you will have to remove it or it will die.
Peppermint shrimps are known to control and prey on Aiptasia as well but they are not a guaranteed. Lots of people have success so it's worth a shot. The peppermint shrimp should have no problem mixing with a cleaner shrimp.
If you can remove the rock, I would take it out and use a screwdriver to clip out the hole (among with a little bit of LR) where the Aiptasia is hiding. Or use super glue gel to cover the entire anemone (and the rock), let it cure and put it back in. This will guaranteed 100% removal (unlike certain type of chemical).