I'm struggling a bit to know what this gadget is - you say it stores water, not heats it only as hot water is needed. You also seem to say it stores water in the boiler - not connected to a separate hot water cylinder.
If the above assumptions are correct, how do you know that the lower temperature (that falls during the morning) is not the direct result of usage of the hot water that has been gradually heating up overnight, but which is taking time to reheat, as cold water feeds into the store?
I am convinced that pressure is a red herring. In most domestic systems, the mains pressure is at its LOWEST in the morning (when the whole neighbourhood awakes and wants a shower) and the supply pipes struggle most to cope with the demand. But if this gadget is STORING hot water inside it, not instantaneously heating it on demand, pressure is not a factor.
Gasman may have a better analysis of this situation.