There's no way any PB/designer estimate of the water volume of that menagerie of shapes and angles and steps and slopes will be anywhere close!
Right as they finish the plastering (or the acid washing next day if you're getting pebble), they'll start the fill. Just before they do (ya gotta watch 'em, cause they'll do it fast), you run out to the street and take a picture of your water meter numbers.
You'll probably be the one to have to turn off the hose(s), and Murphy's Law will make that in the middle of the night, of course! When you do, take another snap of the water meter numbers. The difference between the two numbers will be the water volume of your pool!
If you want to know, you can take an extra pic between filling the pool and the spa, so you'd then have the volume of each body separately, and the total for both.
You can improve the accuracy of the numbers by limiting water use while filling: turn off any irrigation systems, minimize showers and flushes, postpone running the dishwasher or washing machine until after the fill, etc. The less water you use, the more accurate the numbers. You can even count flushes and subtract them from the total (like a couple gallons per flush), if you want to be crazy about it. But that's overkill. Within even several hundred gallons will be more than fine for your chemical dosing math. And the water meter can be off by a percentage or two anyway.
Whatever number you come up with using "The Meter Method" will probably be 1000s of gallons closer than your PB's guesstimate, or anything you could come up with by measuring the dimensions.