Hi Sentinel, welcome!
You won't have any trouble getting a Cottonwood Lakes or Cottonwood Pass permit to enter on the 28th. There are some still available online on the
recreation.gov site. And on those trails, 40% of the quota are held back for walk-in requests, so even if they show full, I am sure you can get a walk-in (they become available at 11 AM the day before the entry date).
You also need the Trailcrest Exit permit (trail "JM35" on the website). Those are all reserved for Aug 30, but again, 40% (10 permits) of the quota of 25 are held back for walk-ins. So you should be able to get one if you go in on the 27th, although you would be competing with other walk-ins who may go in earlier for starts at Onion Valley.
Since the exit permit is not a sure thing, I'd go with the idea of being flexible. Plan A would be the above with the exit permit. Plan B might be to trade your dayhike permit for an overnight entering the 27th or 28th on the MMWT (Main Mt Whitney Trail). If you got one to enter the 27th, you could just hike up to Lone Pine Lake and camp the first night, then on to Trail Camp the second. There are lots of cancellations and no-shows on the MMWT, so it is likely you could get one of those permits.
If neither plans above work, Plan C: Keep your current dayhike permit, spend the night at Cottonwood one night (and sometimes two works, since it seems to never be full). Hike to Trail Peak or Cottonwood Lakes two days before your big hike. Move to Whitney Portal, use a walk-in hiker site (between the day-use and overnight parking lots) the night before your hike. Get up at 3 or 4 AM and go for it.
After you go, please report back what works out.