The RF signal from the transmitter is getting picked up by the wiring, and is being interpreted by the GFCI as a current imbalance.
A different make of GFCI might fix the problem. El Cheapo GFCIs often omit "needless" filtering components like bypass capacitors. RF pickup can be a problem with cheap residential wiring materials (romex and plastic boxes), which provide no RF shielding compared to AC/MC cable or conduit and metal boxes.
