Heteractis magnifica are known to climb to the highest point in the tank. To be honest, as far as I know they are the most difficult anemone to keep, and the most demanding. Oftentimes aquarists won't go near them without 400 watt metal halides. They also need high flow.
Carpet anemones aren't necessarily difficult.
Stichodactyla haddoni is often described as an all-round easy anemone, but the other large species are quite difficult. I imagine that a tank properly equipped for those other species could probably support a Ritteri anemone as well.
Another too-often-forgotten issue with anemones is chemical warfare. An established anemone can pretty much take down most corals. Even tough genera such as
Sarcophyton or
Cladiella can be stressed or killed if touched by an anemone, but they can compete chemically, and an anemone that finds itself placed in a tank with established soft corals and anemones can be stressed, causing the anemone to try to leave the area. I suspect this is the cause of death for otherwise healthy anemones placed in otherwise well-equipped tanks.
If the animal stops moving, and appears healthy, it should be left alone in my opinion. Good luck.
PS
Regularly changed (biweekly) activated carbon and heavy protein skimming can render allelopaths (cnidarian secreted toxins; ie, the aformentioned chemical weapons) harmless.