OK, you have an advantage in your location, I think. From the specs, I assume "SoCal" means "Southern California"? If so, then switch difficult child to an Aussie bush shower for a while. I'm actually considering installing one for use after a swim at the beach. I spent some time living with one of these, made by my father, from scrap pieces and spare bits. The house we lived in at the time (up north, on the coast, off the beaten track a bit, fairly wild place) had no running water; all we had was a rainwater tank at the back door, with a tap leading from the tank for use in the kitchen. It was a cold water only tap. We DID have electricity, so we would boil water in the kettle. To wash up after dinner, we would fill the kitchen sink with water from the kettle.
The shower - it was made from a 4 Litre plastic jerry-can shaped bottle (the sort with a moulded in handle). My father had cut a large hole at the top so we could pour the water in easily. He had then attached a shower hose lower down on the side of the plastic bottle, using plumber's tape to give a water-tight seal around the flexible metal hose. On the wall he had two hooks above head-height - the first hook took the full bottle, the second hook, higher up, took the shower head (on the end of the long hose). To shower - we filled the plastic bottle with two kettles of cold water and one kettle of hot, then hung it on its hook. All this hung over the bathtub (which emptied under the house, which was on stilts - you could look through the plug hole and see the ground below and hope nobody was trying to lie under there and peep through - unlikely, the place was miles from anywhere).
So I would strip off, get in the bathtub and reach for the shower head. As soon as the shower head is lowered below the level of water in the bottle, the water flows. But there isn't much of it! So I would have to quickly wet myself all over including my hair (if I was washing my hair) then hook the shower head back up to stop the water flow. Then it was lather time - shampoo the hair, soap up all over, scrub - whatever I had to do. Then rinse - lift down the shower head. Use the water efficiently. If I was careful, I had enough water for a second shampoo and a conditioner treatment as well.
I learned to be very efficient with my water use, because running out halfway through a shampoo was very annoying, I would have to get out, get dry, get dressed and go to the kitchen to fill the kettle.
Camping supply stores have all sorts of camp showers, but the one my father made was very simple and very effective.
I would seriously suggest setting one up outdoors, tropical island-style as a garden feature. And until he can follow a timer, make him use one of these for washing (he can wear swimming trunks for modesty). If nothing else, it will teach him to appreciate water. After all, we are showering with water that is drinking-quality, there are many places in the world where water of that quality is just not available even for drinking. we need to learn to value our resources.
(I also have to admit - I sometimes got fed up with trying to use our bush shower to wash my hair, so I took to going to the beach for my daily swim with my bottle of shampoo; after my swim I would use the cold shower at the beach to wash my hair and use conditioner. No time limit there! But it WAS cold, and totally out in the open, so I always had to keep my swimsuit on.)
The hot water system - husband & I have been chuckling, because easy child 2/difficult child 2 and BF2 were doing this to us, with their shower usage. They would shower together (it doesn't save water like you would think) and would drain our HUGE solar hot water tank, leaving none for anyone else. easy child 2/difficult child 2 has waist-length hair (but so did I when I learnt to use a bush shower) but BF2 has shoulder-length very thick frizzy hair, tight curl. It goes into an Afro if he doesn't slick it down and pull it back hard into a bandanna. Hair that thick just won't wet easily, the water just runs off it. I've seen him at the beach - he emerges from the waves, his hair dripping - then you see his hair emerging DRY form the water, it's bizarre. So to shampoo it, it takes a lot of shampoo and a lot of water. I do get that. But not 400 Litres, guys!
And now - they live in a small one-bedroom flat which has a 20 Litre quick-recovery hot water service. When that 20 Litres is drained, it's got to re-heat. because it's quick recovery, it WILL reheat immediately, it takes about half an hour. Of course, this is far more expensive to run (they haven't had their first power bill yet).
easy child 2/difficult child 2 was wailing at us how to get her lovely, deep, hot baths that she says she NEEDS, she has to run the hot water with no cold, then wait until it reheats (half an hour) then empty the tank again. She has her bath, then before it's completely cold BF2 climbs into the tub (tricky, because it's narrow and he has broad shoulders), somehow wet his hair and then shampoo it with another tankful of water.
I wouldn't recommend getting a quick recovery hot water system. They cost a lot to run because they're draining electricity at peak use times. They're also a blasted nuisance. But they ARE the heater of choice for small apartments.
By using a bush shower, you're not punishing him so much as giving him an opportunity to learn a different perspective on water. I increasingly disagree with punishment. However, natural consequences - THAT I believe in. And using a bush shower could be an interesting experience for all the family. It's also a good way of cleaning up after a muddy session in the garden. If you look around, you could find ways of using nature to help you - paint your water reservoir black, for example, to save having to fill your bottle using the kettle. Just check before use that it's not TOO hot!
Marg