Thank you, folks, for getting me thinking about this.
I just went ahead and picked up a couple of permits for around July 4 -- 6 days/5 nights, enter Kearsarge Pass, exit Whitney Main Trail. Who knows, maybe with this year's lack of snow, the passes will be passable that early in the season for regular hikers who are not carrying mountaineering equipment (maybe I'll bring my new microspikes!). If not, I will probably end up releasing the permits and doing some other fun trip....