It is possible that the breaker on the generator is bad/worn; it wouldn't be my first suspicion to check, but since you've checked most everything else, it might be worth looking into.
I'd also suggest disconnecting the line from the generator to the transfer switch at the generator and (with AC power off all around) measure the resistance with a multimeter between the hot and neutral. It should be pertty high, hundreds of ohms or more, i.e. just whatever resistance the transfer switch control circuitry has in its unswitched state. If it's a few ohms or less, you have a short somewhere and the circuit breaker is doing its job. With a clamp-on ammeter, you could also see what current is flowing before the breaker trips.