I’d think taking the 12V to the screen and a resistor → 5V regulator you’d have what you need… The resistor needs to handle enough watts to handle the voltage drop to the input of the regulator.
Yes, I’d use a buck converter to get 5V from the 12V. The Pi has a resistive LDO regulator to get the more sensitive 3V3 from the 5V. You want a buck converter that can supply enough current.
As @mcdanlj says, you need a buck converter for this, the Pi draws up to 3A, a linear (resistive) regulator isn’t practical at such high currents.
Buck converters are very efficient, in the 95%+ range. So if the Pi draws 3A at 5V (max) the buck converter will draw just over 1.25A at 12V. (15watts in total).
Unless your 12v display needs more than 3,5A I think you will be fine with a 5A 12V PSU plus a buck converter for 5V. I think this is a good solution.