I have a toy quadcopter (syma X5) which I've built onto a custom frame (lighter) and hang my Möbius action camera from for footage.
The x5 has a sister version (x5c) with a removable camera and the wiring is present on my X5.
I have made an adaptor out of micro jst leads to remove the battery from the Möbius and power it direct from the quad's onboard camera supply. I added 2x 100uF caps to act as a reservior., however:
(1) with no caps, the camera works but footage (saved to the SD card) becomes briefly corrupted if he quad goes behind an object. E.g. a tree is between quad and operator.
(2) with the 2 caps, there is a lot of signal noise and the camera shuts down (and looses footage) within a freecycle seconds of flight.
I wonder why it's worse with the caps than without?
It sounds like you are having a problem with the radio RX/TX on the Quad, interfering with the camera.
Also, caps is a good idea to stabilise the supply voltage. I would add 2200-4700 µF low-ESR cap.
When there is a tree between you and the quad, both ends turn op their TX power to keep the link alive, and it corresponds with the behaviour you are seeing.
Wrap the camera in thin copper foil and ground the foil.
I thought it may be the RX board sucking up the power. The camera that is sold as an accessory must have components that guard against the drop and interference. I don't have one to look at.
There is a "supercap" for dashcam use I could try in the camera.