This is configurable to the 'M' button but isn't mentioned in the manual.

This lets you map a camera custon function button the the M button on the gimbal. This can be very handy because you can in turn map all sorts of camera functions to the cameras custom function button 1.

In case of the BMPCC 4K, what can I do with it?

on my sony camera this feature is absolutely not working. DJI has a lot of unfinished features. this is the reason why I dont buy the DJI transmission combo with handles

C1 or fn1? Is the "or" depending on camera model or is there a real choice between two options? I can't find any other item in the gimbal menu so I guess it should just take whatever "first button" one's specific camera offers? As it stands with the Sony a7s iii, there is no response of either function assigned in-camera when I press the M button on the gimbal. Other functions like Rec Start/Stop or assignments to the gimbal's front dial like ISO and aperture do work. So please can someone help here?
EDIT: It does work via bluetooth but not when using PC remote (USB-C cable attached). But things like ISO/Aperture etc. are only controllable via USB-C. WHY?

Same here! Via usb no c1/fn1 mapping possible, with bluetooth yes (but not the rest)

Seema to be a bug....please DJI, check this!

Thx