Fig leaves and figs in water. Critters and birds attracted to figs. Chlorine water doesn't help figs. Some people find fig trees to smell disgusting i.e. won't want to sit in pool for long if affected.

I think the benefits outweigh the risks. The fig tree will be dormant when the pool is, any problems can be handled then. Ripe fruit can be picked from the water, the tree will have additional reflected light from the pool so it should grow quickly. Get a breed of fig that makes monster leaves to shade you and the ice bucket of Corona and limes. If bugs become a problem, just encourage bats to your pool and let them eat the bugs. It will also help with enforcing the pool hours - when dusk comes so too will hungry bats.

What you are missing here is figs have destructive roots. They can lift a building! They will lift the concrete, and possibly invade your pool.

roots would be a main concern.

how far away ?

i would plant at least 20ft away...

think of shade (suns angle when it grows) and leaves going in the pool also.