The Whitney Main Trail is a lot longer than the Shasta approach, so it is not really easier. If you carry overnight gear up to Trail Camp, that is lots higher than the overnight spot on Shasta, so that part is tougher, too. AMS is a bigger threat there, too. If you camp at Outpost Camp, then that compares more to the Shasta overnight at Helen Lake, but quite a few miles more to the Whitney summit than Shasta. If I were you, I'd stop at Outpost for the overnight. You could even spend two nights there and hike out and drive home the 7th.
Permits: When you go to the Visitor Center in Lone Pine to pick up your overnight July 6 permit, there will very likely be some no-show or cancellations for July 5 available. If there aren't any available, come back 30 minutes later and try again, as they show up as others picking up permits drop off slots. At 11 AM on July 5, all the no-shows for entering that day become available, but since you have a July 6 reservation, the person giving you a permit might be able to assign one to you before 11. If you are there AFTER 11, then all the no-shows will be in their system available to anyone to pick up.
Not sure of your work time-off situation, but if you have July 4 off, too, you MIGHT even walk in and get an overnight permit starting that day. There have been unused ones available on July 4 on previous years.
If switching permit dates makes it so you don't get a parking pass, you can park along the road just down from the hiker parking lot. Roadside parking is available through July 10.
Good luck!
...and please report back how things work out with the permits! (But report in the REAL current conditions thread, or in the Trip Reports forum.) This is the "out-of-date" thread.
