ah.
well, with a damaged lcd you're going to toss after removing, it's much simpler if you just cut the pins between the lcd and the mcu board.
then desolder each pin individually, which shouldnt' take much heat or much effort.
once that's done, and the holes are clean enough to put the new lcd's pins thru, then solder each pin individually, which should work fine too.
alternately if you need the header itself, you can desolder it from the lcd side, because you're replacing the lcd, so it doesnt' matter what you do to that part.

just ensure whatever heat you use doesnt' affect the mcu board.
since the most likely failure is the mcu itself, you could ask grin tech if they sell just the programmed mcu--if so you could replace the one on the dead ca and see if it works now.
the other likely part failure is the pass transistor, which you can get from various electronics places. it's a different type of fet than usual, a depletion-mode, iirc (justin_le posted about it in one of the ca repair threads, possibly mine). if it is the pass transistor, then you could just try powering the ca from a 12v-15v source into the aux-power-input pin (cant' remember it's name, but it's listed on the pcb layout pic on teh ca info page).