Looking at the current No Show deadlines:
Scenarios:

Day use: 12 Noon before Entry Date:

Grab permit 1PM-2PM. Call to setup up the Permit Kiosk. Drive up from sea level. Start hiking at 12 midnight. Start puking from altitude sickness probably around Lone Pine Lake make hasty retreat back to car....

MultiDay (overnight): 10 AM day of Permit:

Grab permit 11AM-12PM. Call to setup the Kiosk. Drive up from Sea level. Sleep at increasing elevation campsites (Lone Pine Lake, Outpost, Trail Camp, etc.) Success!

Can you sleep at Whitney Portal and sneak in the next day? E.g. permit for Thursday-Monday (Monday just in case you still feel sick at Trail Camp and need another day of acclimatization) but sleep Thursday night at the Portal and have breakfast at the Portal Resturaunt.