> it should stay hot for about 24 hours if it is well insulated.
It takes 50 kWh to heat a 46 gallon tank up to 140 F. That's a ton of energy. Hotter, larger tanks lose even more energy.
Instead, get a tankless heater, backing a small heat pump water tank. You get water as hot as you can possibly want, heated whenever you want, and it never goes cold.
Also, water consistency isn't that big a problem. Today's 120v heat pump water heaters store water at higher temperatures and using a mixing valve to deliver the desired temperature of water. I am just saying, surely electric water tanks, mixing valves, and temperature sensors are orders of magnitude cheaper than the amount of batteries needed to heat an equivilant amount of water.
My parents 1988 switched their hot water to a plan where it was all heated at night. 6 people making no effort to conserve water ran out twice in all those years. Yes the.water was hotter in the morning. But the last shower of the day still needed to be mixed with some cold or it was too hot.
Tankless is worse than a large tank. Tankless needs a lot more energy now, less over the full day, but when you turn the water on it needs a lot now. A tank easily adjusts to use power when the power company needs it. Sun shining or wind blowing, then heat water to use up whatever is extra. Clouds and no wind, just use the stored energy.
Maybe a battery is more efficient, but tanks are cheap
I think you missed that the proposal was a tankless heater backed by an air-source electric heat pump heated tank. The idea is that the heat pump may, or may not, get the water to the desired temperature; the on-demand heater just makes up the difference.