I was wondering if the psone lcd and the GC could be powered off of a GC power supply. I haven't been able to find any answer to this yet.

Any input helps.

Yes.

Do I need a step down regulator to get the right voltage for the screen?

well, yes if your using a psone screen, it needs 7.5v everytime to power,
if your planing on using a larger battery than 7.5v, you must use a Adjustable regulator, which you can adjust the output v to the required 7.5v.
your psone screen will receive 7.5v from your large battery everytime it turns on,
if you don't do this, using more than 7.5v power without an adjustable regulator to change your 12.0v or 14.8v etc to 7.5v,
your screen will fry in no more than a mintue.

with the Adjustable regulator the output will be 7.5v your screen will receive each time the required power it needs and there's no risk of it every frying
you only have to adjust it to 7.5v once and your good