Wire the 30A cable up (assuming this is RV-standard 120V 30A and not 120/240V 30A four wire service) so that the 30A hot feeds both hots of the trailer system. That's the way the adapters work, and that way you won't have to worry about which appliances etc. will work in your trailer under generator power. That also means you can use your new cable with a 30A pedestal if needed for some reason (assuming of course it's a TT-30 plug you have for the generator end).
While in motion, it seems to me the chance of CO buildup from the generator, regardless of the fuel source, is extremely low. There's simply too much airflow blowing the fumes away. When parked it is a valid concern to consider and take appropriate steps to ensure safety. Personally, I would probably use the generator on gasoline while on the road, mainly because (at least around here) it's less expensive and a little more handily available.
