PIC Core? it has a regulator and rectifier so it is pretty tolerant. Excess voltage is burned off as heat, so it can be worth trying to supply the minimum you need. I have a 7V AC transformer connected to mine.
But if your PSU can't supply the current needed for the backlight then it won't light up.
I would check the A and K connections of the LCD with your multimeter in continuity mode, being an LED it will pass current one way and not the other way. But if you get no reading both ways then the LED may have popped.
You should get the backlight at the very least as that's a simple circuit.