Quick and easy.
Use a LM7812 based regulator just for the screen. Being linear regulators, 7812 regulators are not very efficient, but will get the job done dirty cheap. Don't forget to heat sink the 7812 chip. Small downside is that when the car is not started, ie when battery voltage is ~12V, 7812 will output only ~10.5V. See if your screen is tolerant to that low voltage. In general, slightly lower voltage should not lead to any problems. When the car is running, the battery voltage goes up to 14.4V and 7812 will regulate the output to solid 12V as required by the screen. Since the voltage differential is not that high, the heat dissipation should not be high either.
IIRC, LM317 is better in terms of drop-out voltage than LM7812 and should help the above idle situation better. LM317 being an adjustable linear regulator, requires 2 resistors to set the output voltage though.
PS: All above assuming that your screen takes less than or equal to 1A current.
Powering 7812/317 from the 19V rail would solve the idle power issue completely, but it would be too inefficient again.
See what suits your situation better.