Pictures of my LC III recap job. 

Through judicious use of freeze spray I've pinpointed the problem to the large "VLSI" chip in the centre of the board - labelled 343S1065 - apparently it's the "Senora" chip, responsible for memory, I/O, audio and video. I don't have a replacement so I'm just gonna try reflowing it when I can get my hands on a hot air tool.