How does the camera button work when recording video? (I looked at the LX3 Manual description, but I'm checking to confirm).
By that, I'm asking the sequence of button pushes to start, stop and then restart the video capture process.
If alec_t is correct (which I think he is), i.e., max 2G file size, but with multiples of that file size allowed as separate, distinct files, up to max the memory card can handle, then maybe a timed, external, solenoid actuated "button pusher" rig might suffice.
BTW, of all the things I've had occasion to disassemble, cameras, IMHO, (especially film - springs, ya know) are often the toughest. Assembly ain't no joy, either. And the PCBs often require a microscope to do any soldering.
Which is why I'm thinking an external solution.