Marla,
Before your PB starts digging, here's one thing to consider w a few confirmations.
You may receive a pass on full sun hitting your pool most of the day, but since we both live in Hou., we know the best times to use our pools is when it's the hottest. From July-Sept, pool water temp's in Hou can really heat-up. W your pool being on the smaller side, and prob 10-15% of it real shallow w tanning shelf and ledge, you might want to consider a chiller. The smaller your pool is, the quicker it will heat-up, and pool temps of 85-88 degrees are going to be the norm if you get 4-6 hrs of sun. Our ambient air in Hou in the summer is sometimes 80+ degrees in the coolest of the mornings. Now, since your pool is smaller, the good news is that it can be chilled down faster. Believe me, I've lived here 48 yrs and getting cooled-off is what it's all about. If you installed a chiller, you will be using it A LOT MORE than you will your fire-pit in Houston. Run that by your PB and the folks here w pools in the Texas areas and see what they think.
Good luck and I wish I could have relocated your tree to our ranch near Columbus TX. Shade in the summer is always appreciated.