I would recommend taking the slightly longer Bull of the Woods trail, at least on the way up. It's prettier and you stay on the ridgeline longer. This means getting a crack of dawn start so you don't have to worry about t-storms.
The last time up there we intended to camp a couple miles in on the trail, but it was raining and we were wussies. The good news is that during June, off-season, you can pop into a couple of the ski lodges and haggle out a reasonable hotel room on the spot. To do a third time, that would be the plan: come ready to backpack it, and then if the weather sucks just enjoy a lodge room instead.
Also, if by any chance you're up there on a Friday and you like (or might end up liking) classical music, stop on by the old grandaddy of the ski area, the Hotel St. Benard. For years they've hosted one of the VERY top college-age music student summer events, and they have concert dinners that are really unique and worth it. Good food, the original ski lodge, and musicians who will likely be the next Yo Yo Ma's etc.