We don't question whether bathbtubs or kitchens or baths should be GLA because their use is critical to living spaces and virtually all buyers want and need the features within a bath or kitchen.
It is all about demand, function, and market acceptance. A 3 bedroom 2 bath home with one kitchen has proven market acceptance and strong buyer demand, a 3 bedroom, 20 bath home with 4 kitchens is...odd.
A living room with a pool is a custom feature. It could have unique, if limited appeal if it is in an area with unique luxury homes with exotic features, or it could have appeal mainly to those who built it. The OP has to take into account market area.
If I were appraising it (I never had one like this,) I would probably include the area, but then would comment on how it might affect marketability, and possibly deduct a bit for loss of function of what most would consider usable living room space. But it would all depend on the layout. If the pool area took up a certain amount of room, but was compensated for with a lot of open living area near pool , or with alternate rooms such as den or family room, liveability might not be negatively affected .
As to whether a well planned indoor pool if maintained well would present mildew or other problems, don't have experience with them, the pools in my area are all outdoor.
Good luck, sounds like a fun (not) assignment!