To me, your diagnosis of the 'problem' that concludes in your mind that you want to install a pump is not correct.
You seem to be saying the temperature reduces (during the day) when the mains water pressure reduces. Generally the reverse would be expected to occur. If this is a mains-pressure supplied water heater, a cooler temperature is received at the output when the pressure is higher (because the same amount of the gas energy is being expended to heat a greater volume of water passing through. Please explain further if you want more advice.