Many years ago, my hiking buddy and I started out on a similar trek. Short story: We bailed after the 4th night -- windiest night ever in the Sierra.
We had really heavy packs, more than 50 lbs. We started up the trail using headlamps, and camped ON the trail about midnight. Then on up to Shepherd Pass the next day. We climbed both Tyndall and Willi on day 3 -- that was nuts! The night at Lake Helen was torture with the wind blowing like a hurricane. Long day out--our car was at Whitney Portal, so we walked the dirt road clear to Onion Valley road to hitch-hike back in the dark.
Anyway, I can't imagine you would encounter any snow or ice requiring crampons this late in the summer. Maybe on Shepherd Pass, but you should be able to walk up it mid-day.
Take it slow and steady, and don't overload your pack! I'm hoping you can do it. Good luck!
