What is causing a motor to trip my GCFI on every OTHER start?
Well, it's not so much every other time as it is every second time.  This suggests that something inside, be it a cap or something else, is storing charge  which back-EMFs the second time you start it up.  Then the breaker trips, everything dumps, and you start clean.
If you can, see if you can :

run the machine, then turn off
use a grounding stick to discharge the cap
turn on again

I know from my days messing with CRT televisions that caps can hold charge for days.