There is nothing wrong with monthly waterchanges, at least not something wrong enough that you can blame the death of your anemone on it. However, I feel that allelopathy is at least partly to blame for what you're experiencing. More frequent larger waterchanges could certainly cut down on the concerns of allelopathy but you're not really fixing the problem.
When it comes down to it, most clown anemones have a horrible survival rate. For the ones that do make it past the retail stage, by the time they reach your aquarium they're pretty stressed. Then people go throwing them in mixed reefs with other anemones and corals. So not only are they fighting to survive from the rough journey they've had, but they have to contend with a toxic soup from other competing animals. The anemone is already expending a lot of energy to recover, but now it has to expend energy to fight off toxic chemicals and produce it's own. Often this will lead to a quick demise or it could take several months to "thin out the herd" of the weaker animals.
Ideally, anemones should be kept in a species tank and just about anything besides a BTA is a big no no in a mixed reef IMO. Anemones live for hundreds of years in the ocean and their reproduction rate is rather low, so providing them with a good environment is important. A mixed reef with another anemone in it is not a good environment for a recently collected carpet anemone.
My advice to you is to give up on your pursuit to keep one of these carpets, at least in your current tank. Though I hope the one that you have pulls through, it doesn't sound good.