While waiting for some more insightful information from you guys, I made some basic measurements of the inverter board and input connector pinout myself. In case this information might be useful to someone else, I'm posting it here. Likely, the pin assignment is as follows:
 
01 Ground
02 Ground
03 Brightness potentiometer out (0–5.15 V)
04 Brightness control from the Power Manager (0.22–5.00 V – probably PWM, but not sure)
05 Unused
06 Power in (7.5 V)
07 Power in (7.5 V)
08 Brightness potentiometer in (5.15 V)
09 Contrast potentiometer in (FSTN displays)
10 Contrast potentiometer out (FSTN displays)
 
The question is if the input voltage is enough to drive a modern LED board, as modern inverter board input voltages seem to be higher. As suggested in the thread at MacRumors, I'll send a mail to lcdparts.net to see if they have anything to suggest.
 
If any of you have any other suggestions, or could confirm or elaborate on the connector measurements (in particular the brightness control signal), it would be very helpful.