I'm a little confused. You say that you can't fit a long heating element but then say that the tank heats slower with a long element. Can you fit a longer element or not? Or do you just not want to?
The short element can heat the whole tank but it takes longer. Your right, a long element would be the best solution.
If you can't or won't fit a longer element then I suggest adding a pump and recirculating the water from the top of the tank to the bottom, or vice versa. Only having hot water in the top of the tank would not be acceptable to me. Regardless of where the heating power was generated or how it was controlled. It sounds to me like this is a more immediate problem than tinkering with the control circuitry.
There is nowhere to fit a long element on the tank. There is only an external short element and it's fitted level with the bottom of the tank, that's why I can't access the thermostat. Maybe it was installed upside down?
It was never a really a problem because we always heated the water with an oil burner. The immersion heater was never used until we had solar PV installed.
A pump sounds like a good solution. Even though the immersion heater is at the bottom of the tank, the top can be 80C and half way down is cold. Still resisting getting plumbing done.