A shower chair is a fast pack as long as it is bone dry first; the only real risk is trapping moisture in a bag and inviting mildew or rust before you even arrive.
Pro tip: A shower chair is a good candidate to keep out and usable until the last possible day, since it is needed daily and packs up in minutes.
Watch out: Never bag a shower chair while it is still wet; trapped moisture in an enclosed bag or box is exactly the environment mold and rust need to take hold.