In the interest of reference I'm attaching pictures taken with a phone camera so that other people know what they are getting themselves into.
I'm not really good with diagnostics but I have taken the whole thing apart and started probing for voltages. Voltage at the control buttons was measured at ~1V, which was quite far from the 3.3V I was expecting.
The problem turned out to be a blown SMD fuse, marked as 0F0 (or similar), rated to 2Amps, which I've replaced with a jumper wire rated at roughly the same amperage.
Interesting fact: the beamer case is painted with some metallic paint which is enough to conduct from about ~400V potential. I found out this the hard way by letting the lamp connectors rest on the case while testing. It produced a huge spark and the paint was stripped out for about 1sq cm at the two points of connection.
View from above with case tken off:
Mainboard view from below:
Another enlargement. The culprits were the small black things in line with the first connector. One of those was tested interrupted and subsequently replaced with a bent single-core wire.
Lamp housing, fresnel lens at the bottom:
Lens housing, 3xLCD connectors, vga signal (?) pcb:
Power supply housing at the left, isolated from everything:
Details of main power supply:
Details of the lamp supply:
Bulb housing from above, the round thing with the thick wires is a temperature sensor for the lamp:
Sorry about the quality of the pictures and the explanations, this was done one year ago and I did not remember to take detailed pictures and mark down the measurements.