Very hard to say without seeing, pics would be good.
Poinciana wood is soft and does decay rapidly, although a wound paint may seems to be the right way to go overall I dont feel it will may a big difference.
I think the one you mention is marketed as Vinevax dressing.
Here's a link.
http://www.ecogrowth.com.au/Vinevax%...%20maanual.pdf
The roots that get cut I dont hold much hope for even if treated.
Really, if this is a large old tree then the fight will be to provide ideal conditions for what remains so the tree can compartmentalize the damage and try to wall of the decay. That means also preparing the tree prior to the cutting, building it's energy levels up.
After cutting a good care program, ensuring mulch, organic nutrients (balanced) and regular water, the tree needs to grow reaction/woundwood but not vigourous new shoots from over fertilizing with nitrogen etc.
Do also realise that if the neighbour is digging a hole for the pool the water/soil moisture level will deplete in the upper layer too, so preparation is vital, preventing the soil drying out where they dig is important too. Then if a concrete pool is going in it could be wise to have some protective sheeting or even plastic between the cut and the concrete.
So imagine they get the usual backhoe or excavator in, usuallly they rip and tear roots, leave the cut face open, pour concrete, backfill.
You'd be wise to cut any roots clean which are over say 10mm dia, keep a soaker hose along that cut and keep it moist, drape something over the face, even green plastic.