My building does not have a pool and is in an area where the local council maintains several harbourside tidal pools but no in-ground facility. Dogs are prohibited at all the council-controlled swimming areas.
Your correspondent’s local council may operate a swimming centre and, if so, its policy on dogs (almost certainly a ban) should provide support in pressing the case for keeping dogs outside the child-proof fence around the pool.
It should not be too hard to create a by-law prohibiting dogs from being brought poolside, even if they are allowed off-leash in the general common area. Such a by-law could be defended as being in line with generally accepted community standards.