Oh boy - what a can of worms.
Firstly, 240 volt appliances and motors do not need a neutral wire because the load is line to line. The neutral is the center tap of the transformer or generator winding, and it is only used for dual voltage systems (120/240 split phase).
Do NOT change any of the wiring in your home. It is wired to code (hopefully) and neutral should be bonded to ground at the service entrance. That is the only neutral/ground bond allowed in the system, including at the generator. The generator should have a floating neutral. To check to see if it has a floating neutral measure the resistance with an ohm meter between a neutral blade in a 120 volt outlet (the neutral is the wider of the two blade holes). There should not be continuity to the ground pin hole (or the generator frame).
This is one reason why portable generators should NOT be used for home standby power - OHSA regulations require them to have a bonded neutral with GFCI protection on 120 volt outlets. If you do have a generator with a bonded neutral, then it will either have to be defeated (refer to the manufacturer's wiring diagram to determine where to disconnect it), or use transfer gear with a switched neutral.
It's not Reliant's fault - it's your fault for buying a portable generator to do the job of a stationary standby unit.
Now that's covered, having two neutral/ground bonds only causes current to flow in the ground, and it may cause GFCI outlets to trip. But it will not cause the main breaker to trip on the generator. You have something hooked up wrong that is dead shorting it if it's the main breaker that's tripping on the gen. Review all your wiring to find the fault.
If it's the GFCI that's tripping on the gen, then it's the dual neutral/ground bond issue and the GFCI is seeing different current in the hot vs the neutral, and tripping it. But that would only be on a 120 volt outlet, and I assume you are plugging into 240.
The third thing that will trip it is overload. Do you accidentally have a large current draw device on when you do the transfer? Remember that a small generator will not power much. And also remember that on 240 you can overload one leg and easily trip the breaker, while that won't happen on grid power. If you have a 5 kW rated generator, it's actual output will be about 4.1 kVA and it will only maintain that for about an hour before the generator windings get hot and the voltage drops below 220. It will maintain 2.5 kVA continuous and maintain proper voltage and frequency. But at 2.5 kVA you only have 5 amps available on each 120 volt leg, and at the full rated 5 kW you only have 10 amps available on each 120 volt leg for initial starting of motors like in your 'fridge.
Standby generators are only rated to run at 50% load continuous. Portables are usually worse than that and some of them will only maintain 30% load continuous. More expensive portables (like Honda) have a 1.0 Power Factor and will develop full rated output power continuous. Generac portables (GP-series) are one of the cheap Chinese ones with a very poor Power Factor so it takes twice the size in a GP-series Generac of what it takes with a Honda EM-X series to run the same load.
When it comes to generators, you get what you pay for. And very likely you bought a generator that's way too small for the job.
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Chris