Admir, your plan is pretty aggressive. Don't be to quick, given your "few that struggled". By the way, everyone notices the shortness of breath, that's due to the diminished oxygen availability. The AMS symptoms you need to watch for are headache, nausea, lethargy, stumbling, etc, etc.
So you know, many people day hike Whitney, and leaving early from Outpost is easier than that since you have a 2k elevation start. Moving from Outpost to Trail Camp is almost a silly thing to do, since you are climbing another 2k in maybe 3 miles. (Of course from the acclimatization standpoint, it's a good thing to do.) Just start later from Outpost, as in after lunch.
On your travel day, (are you coming into Las Vegas?) I would drive all the way up to Whitney Portal, or 30 minutes longer--Horseshoe Meadows. That first night as high as you can get will give you the best head start on acclimating. At Whitney Portal, there are always walk-in overnight sites... see: Backpacker and Walk-in camp sites and locations at Whitney Portal:
Sleeping in your car at Whitney PortalHorseshoe Meadows is a one-night-only area, so sites are always available. It's wide-open, so if it appears full, just find a spot with bare flat ground, and throw down your bedding.
I like bobpickering's line:
"the best sleeping altitude is the highest altitude at which you don’t develop symptoms"
