If you purchased a field-grown, B&B (balled & burlapped) replacement for that “Bloodgood”, it’d be an easy $8,000 - $10,000 just for the tree, not including the $4,000 labor/machinery to dig/transplant/install. Once your pool guy starts digging the pool, he’s gonna hit all kinds of tree roots from it and others.
Make sure they check for utilities, too, by calling VA Miss Utility 811:
A solid, competent contractor always does that before digging anything around residential/commercial buildings. They should have an account with them, and all utilities will be marked with different color spray paints, to warn the diggers.