If the generator is a portable generator, then the ground is almost certainly not bonded to the neutral at the generator, but floating. A neon test light between hot and a floating wire can light dimly--for that matter, one between hot and your fingers will work fine. The three light outlet testers are usually nothing more than three neon test lights, internally, arranged one between each pair of wires on the outlet.
What you're observing is not a fault for a portable generator, but rather an intentional design feature, and would not cause the GFCI to trip.
Troubleshooting would be simpler if you plugged into normal utility power, where ground is bonded to neutral. A hot to ground short there would generally cause a breaker or GFCI to trip (a GFCI that you're plugged into, not necessarily the one in the trailer if the fault is not downstream of it).
