How does the adiabatic cooling of wingtip vortices work?
The vortex is rotating. This by itself should suffice to explain the expansion at its core. The mechanism is less Bernoulli (which applies to straight flows) but centrifugal force. Compare it to the decreasing height at the center of the bathtub vortex when water flows out the drain and starts to rotate. The height is proportional to the pressure in the fluid. The same happens with air - the pressure change is highest at the core because the angular velocity decreases with the distance from the core.

And, yes, air is compressible below Mach 1. It is only incompressible ideally and at Mach 0. The density changes are minuscule at low speed, but they are there. The thermodynamic process is an adiabatic one (no heat flow) with changes to pressure, volume and density (a tiny bit).