This situation could have been better considered at planning stage...before the pool was approved. Im sure that even a one or two metres further offset would have been better for the tree...and in the longer term for the maintenance of the pool itself (and even its structural integity).
Unfortunately, and all too common the Arborist is called upon to resolve these issues TOO LATE. (We are between a rock and a hard place...and extremely difficult to find that middle ground .... where there is a good outcome).
We are all assuming a worst case scenario here (probably because that what we nearly always have to deal with). In fact I held back on any comment on this case because I just could not see....even with photos.... what distance the tree is from the boundary. However, the advice provided so far (by others) is the best that can be provided under the circumstances.
This situation is something we will see over and over...(new) allotment sizes are so small that the only trees with scope to retain are those located at the boundaries. Futhermore...that is the only area where additional trees can be planted...and things can get very tight if the neighbours wishes undertake additional development within their properties.
There IS a trend for pool installation companies to ignore trees (learned experience)...down here they take a substantial deposit before it's even been approved at Council.....and woe betide the proponent if there are problems. So much are their contracts binding that there is little flexibility to even change the pool shape.
It is only through alert Consent Authorities that trees have any hope of being considered.
Consultant Arborists....deliver a handful of business cards to all the pool companies...that is where the whole problem starts.