So, we have not done a split stay like that, but we have done some other hotel hopping, including a road trip to Six flags over Georgia, continuing to KSC, then continuing to Disney. As others have said, we packed multiple bags with the family clothes for the night/next day.
Also, if driving, I recommend packing in the storage tubs. I literally have five that I purchased specifically because they fit in my Honda Pilot. Not only does it make packing, and perhaps loading and unloading easier, they also seem to draw less attention than a car with a bunch of suitcases visible in the back. They are not a "thick" as a suitcase, so you can pack more into less.
And again, you can pack one for each stay, and a carry on type bag for the toiletries.
The ones I have are also the perfect height for my car, so as to not obscure my vision...