We are dealing with the same thing; two trees we don't want to lose. One of them has alot of sentimental value; it was a gift to my mother 10 years ago and she is deceased. Anyway, what I have been told is that the root system comes out under ground about as far as the longest branches above ground. I don't know if that is true or not. Also, that the concrete above the roots might inhibit water from getting to the roots. I think what you have described sounds pretty tight.
We have talked to a tree man who can move our trees and he is very reasonable. He quoted us about $500 for the two trees, but that is sight unseen. His quote might change once he has actually come out and looked at them. However, one of the trees has had a hard time of it this year because of the severe freeze early in the spring. He told us he would not try to move it this year because it has already been shocked once and to move it will shock it again and he does not think it would survive. So looks like if we want to try to save this tree, we will have to wait until next spring.
I just offer this for what it is worth.
Gloria-KC

Another thing I know can kill the trees is when heavy equipment run over the soil nearby the tree.....ie, bobcats,dump trucks. It compacts the soil and is just as tramatic for the tree. That is why home builders usually clear the lot because it is too hard to save a couple of key trees. Also, I am not sure about oaks, but some tree roots are invasive and can grow into the plumbing or crack your pool.....again, not sure which tree's you need to be leary of.

I have a medium-sized live oak that's about six feet from a three foot deck that abuts the negative edge trough. The tree overhangs that deck and the trough; its roots were certainly cut on that side and there was also a good bit of soil added on top of part of the root system. The pool was put in about 20 months ago and the tree is doing fine.

Once word of caution, the acorns will definitely stain the plaster if left on the bottom for a couple of days. The acorns only get into the trough so it's not a big deal. I just dump a jug of bleach in there and that oxidizes the stains in a day or two. If the acorns we're falling in the pool it would be a bigger problem.

The problem might be as the trees grow. We had a huge laurel oak tree about three feet from our house and it had started to lift the foundation of the concrete porch. I'd worry about the root systems more a few years down the road-- they might be small enough now that it wouldn't matter, but later it might cause some trouble.

I have 2 oaks about 5 feet from my spa. Don't do it that way because the vacuum doesn't reach into the spa ;-). The trees have survived, beautifully in fact (3.5 years so far). The just dump a mess of acorns & leaves in it... I love how they overlook the spa when we're sitting in it, but it may have been smarter to put the spa away from the trees...

Around here, we have oaks all over the place and the building departments treat them like some sort of holy shrine.
The biggest thing is just to keep out of their driplines (the area of ground underneath the circumference of the trees' canopies). This is where the roots are, so we cannot dig or alter the grade in these areas.