I tow my Jayco hybrid with a 2012 Chevy Silverado with the 4.8 V8 and 3.23 gears. It does just fine in Florida where it's flat. The owners manual says I can gain 2000 lbs. towing capacity changing to ...
OP, do you have the 6L80 tranny? If so I stand by my previous post assuming you can remember to manually shift out of more than the first OD as needed. Even if you have the 4L60 I would try it before changing as you are in the SE, not the West. Campers have a way of growing and you will be looking for a larger TV sooner than you might think.