Low snow, Sierra loop for 6/1-6/3?

Mineral King wouldn't be an issue for permits, it's much less visited than anywhere else in the park. But the trailheads are at 7800 feet, so you are definitley going to run into snow. Every direction is up, up, and up.

Here is a webcam for the Mineral King valley, I think it updates twice a day.
http://www.mk-webcam.net/

If the answer is, "not too high", there are some nice options in the Golden Trout Wilderness. One that comes to mind is a Doe Meadow and Kern Flat Loop. Leave from the Jerkey Meadow trailhead and follow the Jordon Trail down and across (via bridge) the Little Kern. Continue past the Trout Meadow Ranger Station and drop down (and down) into the main Kern River canyon heading to Kern Flat. From there follow the main Kern River downstream to the Forks of the Kern and head back up (and up) reaching the Forks of the Kern trailhead. From there you can stroll the final 2 miles along forest road(s) back to the Jerkey Meadow trailhead. Total mileage for this loop is 26.8 miles with some up and down. Elevations will run from ~7,000 ft at the high points down to ~4,800ft at the Forks of the Kern. The only tricky part might be the final ford across the Little Kern (at the Forks) which can be a bit tough early in the season. The flows have already dropped significantly on the main Kern below the Forks so the Little kern should be fordable by now. Permits should be no problem.

I just got out of Sequoia. Traveled up to the Tablelands at 11,000+ft. Snow was never an issue. What was around, was easy to walk across. It was rare at 8500ft, and consistent (depending on aspect) at 9,500ft and present regardless of aspect at 10,500ft. I was able to reach 11,000ft by following rock virtually the entire way.

You might use one of the trailheads out of Jennie Lakes Wilderness.

Thanks a lot for the feedback, much appreciated! I think I'll test out the Golden Trout Wilderness loop. Sounds like it could be done in either 2 or 3 days so the flexibility would be good.