A GFI will not trip due to high current. A GFI will also not trip unless there is voltage to the GFI. A GFI trips when there is even a tiny ground fault.
You have a "Ground Fault". Plugging into the circuit with a GFI (Ground Fault Interrupter) causes the GFI to trip. Plugging into the other circuit which does not have a GFI of course will not trip anything.
It only takes a few milliamps of leakage to trip a GFI. When you unplug the refer from the GFI circuit, you disconnect both hot and neutral wire so GFI does not trip. Fault is in the refer circuits. It could be a heating element leak, corrosion, or other.
I wish you good luck and happy trails ahead!