Yes you could probably accomplice it with a transistor or maybe FET but I think you should reconsider the diode. A quick google came up with
BAS416 Low-leakage diode
The leakage is stated as 3p amps (p = pico = 10^-12) which at 5V is the equivalent of 1 trillion ohms. Your board and wiring are likely to have more leakage (from contamination, no clean solder flux , moisture etc)
And I doubt you will find a transistor or Mosfet with less leakage, or even as low a leakage.
Another suggestion is a low power logic gate (suggestion by another forum member). Look at parts like the SN74AUP1G79 . I didn't read into the specs very far but the front page stated 0.9µA static current.
If you used this type of device you would not have to re-fresh the LCD every minute (it would be on all the time, or off all the time). So take a look at your power consumption and see what you think works best.