How big is your tank? The answer to that will determine what your action should be.
If your tank is large enough to accommodate either a Copperband, Pyramid or Kleini (blue-headed, per
@vetteguy53081) butterflyfish, then my recommendation would be to include one of them in your stocking list. A single aiptasia is not a threat; it's a plague of aiptasia that's a problem. Those butterflies will likely not eliminate all of your aiptasia, but they will keep it in check so that it never develops to a plague.
Peppermint shrimp are problematic; while there are documented examples of them eating aiptasia, not all peppermints will eat aiptasia, and they will also prey on other fleshy corals (or damage corals to steal food from tehm). If you decide to get a peppermint, you should do so because you like them and want them in your tank, not because you expect them to deal with aiptasia.
Berghia are one of the only things guaranteed to eat aiptasia, but they're small, expensive, vulnerable and slow. Given time and the opportunity to breed they will knock down an aiptasia plague, but until you have a plague of aiptasia they're really not the right choice.
Chemical and mechanical solutions are risky when dealing with single aiptasia. Aiptasia are quick to release planula larva when they detect a threat, and all it takes is for one or two of those larva to make it free and grow into new aiptasia. Still, if your tank is too small for the piscine solutions and the invertebrate solutions aren't working for you, this might be your only choice.
If you're going to use a chemical or mechanical solution, best thing to do is to remove the rock that the aiptasia is on and apply the treatment outside of the water. In air, planula larvae can't escape from the parent anemone, and you can more perfectly seal it/scrape it off.
Trying to directly attack aiptasia leads to temporary relief until the new anemones begin to grow. So in short:
Fish-based solutions are best, but you need to carefully select the fish that has the least chance to attack corals. Most fish that will eat aiptasia will also eat corals. Invertebrate solutions are acceptable but hit-and-miss, and you may need to try multiple times to get them to work. Human-based solutions rarely succeed and should really only be attempted outside of the tank.
!
