The storage heaters I have just now are awful, they release the majority of the heat during the day when Im at work and by evening time the house is cold but I do understand modern storage heaters are much better at addressing this issue. So much to think about!
I remember storage heaters from the 1970/80's and they behaved as you describe. I don't know how much better they are now.
If you don't need heat during the day, then maybe radiant panel heaters that come on in the morning and then when you get home might just work?
The theory being the direct heat fired at you from the panel makes you feel warm, even when the fabric of the room isn't.
Personally I'm not convinced, but no doubt others may be.
Although not off-peak, I prefer the 'gentler' heat from the cast convector heaters that run cooler, but are very efficient at getting their heat into the air. We previously had convector heaters with the old spiral element inside that got really hot, and we got a sort of 'burnt' and dry effect in the air that was sometimes unpleasant. But as I say, using day rate electricity is expensive.
I'm not sure, but I think if you changed from E7 to 'normal' rate, and then wanted to change back to E7, you would not be able to.
A thought - how about changing your one failed storage heater for a modern one, and see how you get on with it? Or alternatively change it for a panel if you fancy that option, and see how that goes. Unfortunately you'll need to wait for cold weather to complete the experiment!
There's a risk of your chosen unit not living up to expectations, but at least you won't have bought 4 of them!