Up to now I have been running the camera without any synchronisation to the projector, then manually removing duplicates and blurred frames. This is both tedious and time-consuming!
I have now obtained a cheap Bluetooth AB-Shutter control. I prised it apart with a knife to get to the push-button solder pads. My release has two buttons, one for Android and one for iOS. In practice either button triggers my Android mobile, so I just attached wires to the most convenient.
Once that was done I soldered those wires to some stiff household mains wire. A number of similar projects use a magnet and reed switch to sense frame advance. My cheaper and simpler approach is to use some conductive copper tape on the main pulley and arrange my two wires to be shorted across it once per rotation - aluminium foil would probably work just as well.
The video below shows how it works. I wanted to show how the Bluetooth shutter control flashed for each contact, but since it would only do that whilst paired with my smartphone, I ended up with a series of very short videos!
In operation it works very well, triggering the camera just after the claw has finished pulling down the frame, although I'll admit the timing was more luck than judgement.