Just a couple bits of info......
-Trail Crest Exit (Permits 25 per day)
-Kearsarge Pass Entry (Permits 60, with 36 reserved, 24 walk-in)
-Cottonwood Pass Entry (Permits 40, with 24 reserve, 16 walk-in)
Last year was my 50th year of hiking and have done all the routes several times. The key question for me is, how much time are you allocating for the trip since that will determine route?
Obviously logistics is a major issue. For that matter, I'd skip the HST and do it at a later date. As for entry point, again, how much time?
For me, the Onion Valley to Portal is the most enjoyable and really gives you a excellent feel for the backcountry.
As for timing, last year I had snow on my May 4th trip down to the Kern River. My last trip, end of September, first of October...two days of snow. Both times, nothing significant.
Fall is a great time. Less people, nicer weather, cooler days and nights.
By the way, how many people are going?
paul