While storing the generator on the tongue may work well, I wouldn't run the generator on the tongue. IMHO the exhaust is to close to the trailer and generators with carbs and no Cat converter put out to much CO for me to be comfortable with. I always run my generator at least 10' away from the trailer with the exhaust pointed down wind.
I've had my CO detector go off with someone else running a generator about 20' away from my trailer with the exhaust pointed towards my trailer.
I also NEVER run have a running generator under my trailer for the same reason.