Red squirrel, flying squirrel, bat, bird, are among the possibilities. You can try using a small Havahart trap baited with peanut butter inside the stove - worked for me with a flying squirrel.

If there really is no way an animal could have entered the chimney from the top - and they can squeeze through a miniscule opening - then perhaps it's not in the chimney at all but in the wall itself. This might explain why you hear it only in the fall and spring - possibly attracted to that area by the warmth. If it's been going on for more than a year and the whatever it is survives the stove being used, I'd say it probably is not in the chimney or has access in and out. If this is the case, a professsional exterminator/animal rescuer is your best bet.