Stephen, you are right, the bottom half is the current sensor. It was designed for 3 phase, current running through L1, L2 and L3. DH thinks it may not work on single phase. He gave you the simple way to install it. I can show you how you can try to use the current sensor, but only after the experts show you how to wire it.
Wire it the way DH showed and see if that part of your contactor works. It will be simple to add in the sensor circuit.
Keep in mind there was no current sensor when you had the pressure switch installed. Never seen a small air compressor wired with a current sensor, so what's the big deal if you leave it out? Your 3 phase contactor was an after market add on.