I don't think it's a good idea. It's an extremely active dog park that is literally for dogs off leash to play and run around. If you want to hang out and BBQ I suggest you visit other areas of Balboa Park
I think that might be pretty tough to pull off. There isn't really an area without dogs running around. The park is large but not really designed for grilling and picnics.
Thanks! Your answer is awaiting moderation.