Welcome, awfybig!
The trail head is in the same location as Whitney Portal campground, the campground is just half a mile below the trail head. I've not seen it called Whitney Trailhead campground.
I might modify the itinerary this way:
1. Lone Pine motel.
2. Day 2: Go to the Visitor Center a mile S of Lone Pine and pick up your hiking permit. Drive up to Whitney Portal just to explore. Find the trailhead, visit the portal store, etc. Drive back down almost to Lone Pine, take Horseshoe Meadows road up to Horseshoe Meadows (elev. is 10k', ~1 hour from Whitney Portal). Day hike
to Trail Peak. Consider camping overnight at Horseshoe Meadows, pretty much always spots available. You will get better acclimation sleeping there.
3. Drive to Whitney Portal and start your hike. Trail Camp is a 4,000 elevation change above the trail head, so it is a full day's hike. If you are feeling the AMS, consider camping instead at Outpost Camp (about 10,5k). The summit can be reached from there, just get up earlier. Sleeping there is easier on the AMS symptoms.
Day 4: Hike to the summit, get down, pack out. (It will be easier to pack out if your gear is at Outpost.)
Driving: Stop overnight in Mammoth Lakes. Much closer than Bridgeport, lots of lodging available in summer (it is a big winter ski resort.) Get up early, drive up to the Mammoth Crest overlook (just 15 minutes above Mammoth Lakes), get another spectacular view of the Sierra Nevada -- It is different there than the Whitney View.
Then, drive through Yosemite and Tioga Pass (Hwy 120) rather than farther north. Tioga Pass is a spectacular drive, you should not miss it.
If you have time, (Get up EARLY in Mammoth!) drive down into Yosemite Valley (~2 hours from Mammoth) stopping at the turnouts and snap pictures. It's only 1 hour (round trip) out of your way, but with stops, longer.
Drive the Hwy 120 to San Francisco. ~4 hours from Yosemite Valley.
