Why does my food truck trip GFCI outlets only when a grounded cord is used?
GFCI's work by measuring the current going out on the hot wire and the current coming back on the neutral. If there is more than ~5ma of current difference they trip.

So, if it does not trip when you use the 2 wire cord and then trips with a grounded cord, that tells me your equipment or cord has a problem. If it trips with just the cord with nothing attached to it, then the cord seems to be the problem and you should try another cord.

If a piece of a equipment has a ground fault problem and is leaking current to the ground wire, then a two wire cord continues to work, while a three wire cord allows the current to leak to ground and trips the GFCI.

This is a dangerous situation and should be corrected immediately. That is the purpose of the GFCI. It tells you when your equipment has a problem. Repair it now.

Good luck!