1) The center of the pool is not really where it is critical, it is around the perimeter
2) Any contractor that tells you "the machine is heavy enough to compact it" needs to go. I have had a number of contractors that said things like that ended up removed from the site.
3) A jumping jack is a toy - used only for trenches where it is the only thing that fits
4) A plate compactor that a normal homeowner can rent and move around in their SUV is a bigger toy
5) If your fill is at the wrong moisture content, you can compact it with the biggest thing you can find, and when it gets to the right moisture content it will still settle. Compaction is about understanding and knowing the compaction curves of the particular soil you are working with - not how hard you can hit it.
6) 3/4" minus or 1" minus (not 1/4" minus) is extremely forgiving when it come to moisture content (it has a very flat Proctor curve). It goes by a load of different names across the country - Quarry Process, QP, Crusher Run, Pack, DGA, Dense Graded Aggregate, Shoulder Stone, and there are a bunch of DOT specifications for it that cost more because they are very specific. It is also a bear to shovel by hand. 1/4" minus is normally used as paver base, 3/4" or 1" minus is road base and fill.
For the OP, you should be OK with any clean fill in the center area. I would go with QP. Install it in 6" lifts, and compact it with as large a plate as you can manage to rent. If there is any settling the liner will stretch to accommodate it. Do not use topsoil, other organic fill, and I would stay away from any sort of expansive soils (clearly).