after some minutes looking at the new schematic yet another notes:

- replace 74HC373 with 74HC573: they are the same but have better pinout;
- 74HC138 has three C inputs: /C3, /C2 and C1 (last one is high-level enable input). Check if they are connected correctly;
- some LCD backlights based on LED, sink current above 100mA. So please check datasheet for this parameter.
I recommend you to use ULN2003/2803 after 373; these chips are open-drain ones prepared to control lamps, relays, etc.

Finally, take my another suggestion which frees one more pin (here: P1.6):

- replace upper 74HC373 with 74HC574;
- connect its strobe input (pin 11) to either /Y2 or /Y3.
This way:
- you save P1.6;
- you may not use external pullup resistor and write to 74HC574 just via MOVX like you do for RAMs. That`s quick and easy!

Regards,
Oleg