For instance, there are routers that use a 12vdc wall wart, but, they don’t actually require that voltage to function. In fact, they drop the voltage down farther once power enters the board. Some can run at 9vdc or less.
You should see if your new LCD will actually run at lower voltages before just expecting that you will require the full 12vdc to power it.
**Note: The above is just my opinion and has no solid basis in reality.
Unless you can find some proper tech specs for your LCD display, you’ll just have to try it out yourself as birdmun suggests. See if you can find a variable voltage supply, or use a combination of slightly dead AA’s to get ~11V or less, and see if the display will run from that.