a zip file of a recording of the noise
Very interesting. A whistle starts when the monitor is powered on, and there is a periodic sound when the camera is recording -- so 2 different noises. Maybe there's a bit of hiss when you turn on the video out too.
Is this what you hear?
I would try replacing the 12V switchmode regulator with a linear regulator, but I'm not sure that would be a trivial thing for you to do.
I'm wondering if there is insufficient input voltage and it's misbehaving somehow. The outputs are variable. If the monitor still operates at say, 10.5V, wind the switchmode regulator to 10.5V and see if the whistle is reduced, or changes pitch.
davenn points out that the outputs seem to be mislabelled. Can you tell us what the actual output voltages are for each device, both with the device turned off and again when turned on. e.g. 5D output = 7.25 V with camera off, 7.18V with camera on and recording.