I was thinking of taking a couple of five-gallon buckets with lids and a small plastic wading pool, plus our "privacy pop-up tent".
You put water in the bucket at some point, and set it in the sun at your campsite for a few hours.
You pop up the tent, put the little wading pool in it, and when the water is warm you set that inside the wading pool. Then in you go. Take off whatever; use a ladle, a small plastic pitcher, or even a toy bucket. Sluice yourself jucidiously. Use low-suds detergent/soap/product. Rinse. Dry. Re-dress.
Now. You should still have some warm water, and you have some slightly sudsy water in the wading pool. You can pour the water into an empty bucket and cap it for now, or drag the pool somewhere you can legitimately dump it. If it's too full, you used too much water. Some campsites have a "gravel drain" near the water supply where you can let it run down.
Eventually you'll find a proper place to pour the sudsy water. Repeat with the black bucket of fresh water as needed.
You can haul the buckets wherever in your set-up works best. You don't have to start with a full five gallons. I sure wouldn't. And you can add fresh water as needed.
The wading pool water can also be used for the kids or dogs to play in, same as they might a shallow bathtub...the little bit of suds and dirt will hardly matter compared to what their little feet will drag in there. You can always gently rinse them off with the ladle later.
My over-the-winter thoughts about how to manage a shower where there is no shower, coupled with how incredibly hot a dark-colored, lidded bucket of water got, standing in the sun one day.
You can eventually find all components at yard sales or 2nd hand stores, though we did buy our pop-up privacy tent new, online. It's free-standing, and THAT IS IMPORTANT! Some need a top tether or they sort of slither downward. Make sure your's is self-supporting!
YMMV, just IMHO and all that.