I am pretty sure the first-come-first-serve car-camping sites are good for multi-day. I would figure out which sites they are, then walk around to each one and check the tag on the post (At least the overnight walk-in sites have tags like that). The tag should have the exit date on it, so you can determine when the site becomes available. Since you are going there a day or two before the others, this should work. You could maybe even find one vacating the next day, talk to the people to find when they are leaving, then you could spend a night at Horseshoe Meadows, returning in time to claim the site.
If you don't get a car-camping site:
Your single permit is good to park the night before, so you can use the walk-in sites (just below the trail head). There are always walk-in sites available. There are additional ones below the overflow parking lot.
Here's what I would do with multiple cars: Either the evening before, or the morning you start the hike... All cars drive to the Portal, and unload people and gear. If spending the night before, use the walk-in camping sites. Drive all cars down to the Lone Pine Creek campground and park the extra cars in the long-term spot there. If none available, use the paid spots in town. Or just park along the Horseshoe Meadows road. Then with the one car that can park at the trail head, drive back up to the Portal.
