Alpha Centauri A is not an M-type red dwarf, it is a G2V, something similar to Sol. I don't know Avatar lore, so I'm not sure if you mean Proxima Centauri.

If you're discussing such an exomoon in AC-A, it's definitely feasible, the problems are whether or not a natural satellite can be that large.

If you're discussing Proxima Centauri, then it's still feasible, but less likely. The exomoon will have to be located within the gas giant's magnetosphere, since M-type stars are so active, potentially close enough for a large object like the exomoon to be ripped apart. This wouldn't be an issue with ultra-cool M-types, but PC is not one of them.

I'll have to say though, the plants and other wildlife would be completely inaccurate, especially the color.