The Whitney hike is so arduous that I don't think it is reasonable to add those 4 miles and 2300' elevation gain. The road will be open, but there may be delays up to 30 minutes.
I think you will find parking in the hiker parking lot if you drive up early in the morning on 8/26. Since you are spending several nights on the trail, you don't need to start hiking early, but you should park the car as early as you can.
Your parking permit will then enable you to do the hike from Whitney Portal. I wouldn't consider parking at the lower campground.
