Since even Jewish stores are sued for their dress code in NYC, if the rules of gender separation are ruled discriminatory, even a private pool could run into legal issues here.
Just to be technical, a store is a public accommodation. A private pool belonging to Satmar would not be, so they could invite whomever they want. They could run into trouble if they open the pool to the paying public instead of holding it for the exclusive use of the Satmar sects.