pcator-
I have used my Honda EU 1000i underway on my CD-22 Cruiser, setting it on the cooler which itsef is placed at the back of the cockpit. I use it like this to recharge my batteries after trolling for a few hours or more with my (main) engine mounted trolling motor.
I've not had any trouble with this arrangement, and I've been very careful to note any fumes, though you can't count on smelling them.
I do employ a carbon monoxide detector in the cabin, too.
What one has to be aware of, in addition to all the carbon monoxide dangers, is the "station wagon effect".
This is where an open rear window (door on the boat) pulls in air from the rear of the vehicle because of the lower pressure back there due to the vehicle's shape traveling through the wind.
You have two choices to deal with this:
One is to open all windows and hatches so the flow is into the cabin and then out the back door.
The second is to simply shut the back door to block any incoming air, and then get some positive ventilation with windows and hatches.
Only experimentation will tell you which one works with your boat, but be sure to understand that the airflow through the cabin changes constantly with the angle of the wind's variation to the boat. A light ribbon tell-tale hung in the doorway would be a partial indicator of the airflow there.
I do think it is possible to run the generator with the boat underway, but extreme care must be taken.
Running the generator on the swim step may not be an option due to the sea conditions, and the in-cockpit positioning should have the generator up high enough that the exhaust vent is as higher than the gunnels as well as as near the edges of the boat as feasible as well. My positioning (as noted above) blows the exhaust out over the engine well to help clear it aft.
In your case, you're probably going to be running the generator to power the A/C, so you will have a closed cabin. Be sure to have some positive fresh air inlet and not be sucking air in past the cabin door.
What type of A/C arrangement do you have?
I'm sure some will disagree with parts or all of this!
Joe.