We have been to Gros Ventre a number of times at the beginning of June over the years. The campground has been maybe 10% to 20% full that early. If I remember correctly there are 36 sites in loop D with electric hookups. May 29th this year is Memorial Day so folks that came for the holiday will likely be leaving when you arrive. Kids are still in school so families with kids likely won't be there. Can't guarantee, but IMO you have a very good chance of getting an electric site.
I see that you have 400 watts of solar. Loop A has many sites in full sun, so perhaps you won't need an electric site. The sites on the east side of loop A are a very short walk to the river, which will be running high at that time due to snow melt.
On edit: Here is a link to the Park's info on
Gros Ventre. Be advised that the header picture was not taken at Gros Ventre, as the trees there are nearly all Cottonwoods.