Just posted in your other thread about Gros Ventre, so won't repeat here.
I doubt that you would have any problem getting a site at Colter Bay campground (not the RV Park) at the end of May. Since you are heading up to Yellowstone after a few days, it would save you miles of towing to stay at Colter Bay.
That said, don't miss seeing the Tetons in their glory by driving the TV south to Jackson Lake Junction then turn left over the dam and on to Jenny Lake using the inner Park road. Great views of the mountains driving in that direction! Have lunch on the upper deck at Dornan's just over the river from Moose and then head back north on route 191 to complete the loop.