In fact forget what I said....
If you look at the CTE shield jumpers 3,4 & 5 control the connection to LEDA+. The default for these jumpers in JP3 & JP5 disconnected and JP4 connected. If you desolder JP4 then the shield no longer connects to the LEDA+ pin.
Now look at the LCD and the pin next to LEDA+ is labelled 5v..... This is a 5v connection to the backlight. Just connect your 5v supply to this pin through a current limiting resistor which limits the current to 620mA.
This will supply all the power to the backlight from the alternative supply leaving the Due to simply power the rest of the circuitry which appears to be 100mA
Simples.....