I go to Yellowstone yearly although I do not camp. Here are my thoughts:
Will you be tent camping? The Cooke City area campgrounds are mostly National Forest Land and may not be accessible that early in June depending on how much snow they have this winter. They are mostly east of Cooke City so that is going up in elevation. One of the campgrounds allows ONLY hard sided rigs after a horrible fatal incident in 2010. I do have a friend who is also a Destination Expert on the Trip Advisor Yellowstone forum who I can ask about the Cooke City Campgrounds if you think this is what you want to do.
IF you haven't found this page yet, here is a link to the official Yellowstone NP page giving the open/close dates for the campgrounds. Some are reservable and managed by Xanterra, the lodging concessioner, some are reservable and managed by YNP and booked thru recreation.gov. The National Forest campgrounds are bookable thru recreation.gov
https://www.nps.gov/yell/planyourvisit/campgrounds.htm
I'd probably go with 2 nights at Mammoth which is lower in elevation than some of the other campgrounds across the Northern Range.
2 nights at Madison looks fine for seeing the thermal areas and your 1 night at Bridge Bay is fine too.
I don't like to move hotels so not sure how much trouble it will be for you to pack up but I'd probably do 2 nights at either Jenny Lake or Signal Mountain. I've stayed in the Signal Mountain Lodge a number of times and like that location. The restaurant there is good if you want a meal out. The Jenny Lake Lodge is more upscale than I want to do so have never eaten there.
Depending on the snow load from this winter, hiking may be better in the Northern end of Yellowstone along the road from Mammoth to Lamar Valley as this is lower in elevation than the Canyon/OF area. Still, there will be plenty to do.
This is a primo time of year for wildlife sighting, particularly bears and particularly in the area around the Tower Junction. This is casually referred to as the "Bear-muda Triangle", hahaha. LOTS of Sows with cubs from about the road going back to Hellroaring Trailhead around thru Tower Junction and up to Tower Fall. Both Grizzly bear and Black Bear.
Did you have any hikes you were targeting?