I grew Mulberry trees in Townsville not the weeping ones though their roots were ok.

But I would think again about planting any fruit tree or any tree for that matter that has berries on it near a pool as the wind and the birds can make a bit of a mess I only has palms around my pool there as it was too hot and dry for anything else.

Maybe your local nursery could give you advise on what is the best to plant in that area.