I managed to identify all the large chips on the internal board, I have a list of all of them here
The ones that look important are:
Chrontel CH7034B HDTV/VGA/LVDS Encoder:
This chip appears to be the one that drives the LVDS screen and the VGA output on the lapdock.
explore EP9351 HDMI receiver:
This is the chip that receives the HDMI 1.3 signals from the connected device and converts those signals into something the Chrontel and the audio controller can understand. This chip also has HDCP 1.4 which may be the source of our problems.
Texas Instruments M430F5521 MSP430:
This chip seems to communicate with the explore chip over I2C and the hub over USB, which may also be the source of our problems