In our experience, you will not need to be in any hurry to get to Gros Ventre by 11:00 AM on May 30. That date is the end of the Memorial Day holiday weekend, so the holiday campers will be leaving to go back to work. We have been at Gros Ventre many times over the years and never had any problem getting a site on any day through the end June.
Most recently, we drove over from Lander on Saturday, May 31, 2014 and found only about 20 other sites occupied in the entire campground. We likely arrived late afternoon as we aren't early risers and I know we stopped for an hour or so to explore the waterfall at the Falls Campground near the top of Togwotee Pass.
Take your time - the drive from Dubois over the pass is beautiful.
Also, I can confirm that there is one loop with electric at Gros Ventre.
Upper portion of the waterfall at Falls Campground.
