great point regarding uniformity support on the back fill vs non back fill part. Will talk to my PB to see option of leaving a pump at bottom of deep end.
Travertine pavers are set with very little to no joint spacing so any settling is far more noticeable than with typical concrete pavers. They can also chip if they settle. The concrete slab will give uniform support so individual paver settling is eliminated. Your backfill will have dissimilar materials. The areas on top of the stone backfill will have very little to no movement and all other areas will be subject to movement from frost.
I would ask if he would be willing to do a slab and what the extra cost would be.
Nice to see they are going to run drain to daylight. They should encase the outfall pipe in stone and filter fabric.
This of course will only drain the water to to the bottom of the pool walls. I am not sure if it will be a issue if you have to replace the liner. I would certainly want to be able to lower the water table to the bottom of the pool if I had to change out a liner. Hopefully others will chime in with there opinion.