"Will sliding it under my van be sufficient protection, or will I need a more substantial cover?"
another entry to the Darwin awards !
In no way shape or form put a gas powered generator under a vehicle your in, we had a 43ft long 5th wheel with its own inbuilt onan generator and we nearly died in that too, due to winds circulating the exhaust under the trailer, as it was parked next to a wood pile all along one side.
We had 2 carbon monoxide alarms, bedroom portable went off first then the main trailer one.
The next day we added a 12 ft exhaust extension and put the outlet the other side of the wood pile.
This was in the snow at 8500ft, so a similar situation where we had snow build up around the trailer and the wood pile became a solid wall
Wind direction, snow buildup, position of geny and surrounding snow drifts etc will all effect where the fumes go.
Position down wind, make a small open sided cover out of aluminum or similar position the exhaust and open side down wind of the trailer.
Use while awake, turn off when its time for bed, its a silent killer and you won't be able to smell the fumes...trust me.!