So I assume it is working now?
Ohyes!
To measure this you don't need to use an Arduino.
Just hook your meter between the LCD backlight power pin and power.
That will tell you the maximum current the backlight draws.
Thats what i did. when trying to pwm value 220 it gave me about 10mA.
I did check the LCD and i measured 100ohm from pin 15 to lcd "A"(anode). So no worries about the excess current. Im suprised on how bright the lcd is even how low value resistor it has.