I don't know for sure, but I was told that aiptasia is never gone completely.
Couple things, not all CBB's, regardless of where they come from, are guaranteed to eat aiptasia. I've always trained mine by QT'ing them and feeding them nothing but aip., then after a few weeks released them in to the main system. Some CBB's just prefer fish food and mysis to tiny ugly anemones.
Also, the fish is too big to get ALL the aiptasia, so no- with a CBB you will never full get rid of the stuff. If your CBB eats it though he'll keep it under control and they won't fully infest your tank. But dont forget, this is a BUTTERFLY fish we're talking about so there's a chance it will nip on soft corals, lps and clam mantles. Although a well fed CBB will likely do little more than take an occasional nibble and the coral will be hardly affected.
Peppermint shrimp are also a pretty bad idea... they'll eat aiptasia at first but the second you feed your fish or coral they'll never eat another aiptasia again. They also have horrible mortality rates.
I've found the best (and sadly most expensive) solution is Berghia Nudibranchs. They are small enough to get to every little tiny aiptasia and fully purge your tank. The problems with berghia are they're nudis, so if you have powerheads they will get sucked in eventually... even foamed off inputs will usually have enough intake pressure to kill them. You also need more than 1, I recommend 2 minimum so that they can breed which they do quite easily. If you have a large tank you might think about getting 1 for every 10 to 30 gallons of tank capacity depending on the level of infestation. You also need to keep an eye out for when they've polished off all your aiptasia... they'll start wandering the tank looking for food and eventually starve unless you pass them on to another reefer to use
Just my 2 cents... or like 2 dollars in this case.