Go buy a boost regulator from eBay (they are listed by the hundreds - or at least dozens of the variety you want) and run a 5V LCD screen from it.
The 3V LCD module uses a "charge pump" to double the voltage to the HD44780 equivalent chip, so it is already operating at 3V logic levels with the same chip as the 5V one. The extra voltage is only required to generate the contrast. If the voltage drops to 2.5V, the charge pump is delivering only 5V and you have set your contrast potentiometer in the expectation of 6V, so Vo will be set to around 1.3V giving a contrast voltage of 3.7 instead of 4.7V.
Actually, if you could identify the 5V from the charge pump (it is probably jumper J2 or J3 adjacent to the charge pump U3) of your 3V LCD, you could probably wire a 4.7V Zener from that to Vo and a 2k2 resistor from Vo to ground instead of the potentiometer, to deliver a fixed voltage to the contrast resistor chain. The 4.7V Zener may however be too low (at this low current), perhaps a 5.1V Zener with a 2k2 to ground and a resistor to Vo of the order of 1 to 2k.
The backlight may be more difficult to manage - if you use the option of the boost regulator, you might just have to use it to power the backlight as well.