"Tom, what is Midnight Lake going to be like after July 15 or so?"
Probably a bit damper and buggier, but being on the east side of the crest and more susceptible to the Owens Valley heat, my guess is that the snow won't hang around much longer than usual. I haven't been following the weather down there closely yet, as I'm not going in until August, but if it is a normal June, the snow will melt pretty fast there. The best bet would be to call the White Mountains RS. Or the guys at the permit office. They always seem to know more than the front office types at the RS, IME. Miter Basin is a bit more of a stretch, but still a possibility. Worth a call to the Interagency Visitor Center, as it's real pretty back in there and fits OP's trip requirements nicely.