Here's my lousy plan. 9V battery puts 9V on each of the +sig and -sig incoming signal lines. A transformer makes a new signal out of the difference between +sig and -sig (which should be twice the signal, with noise cancelled out).
The transformer specs are inset at the bottom-right. I have probably really screwed up the impedances for the mic (200 ohms) and the camera (don't know). Please send suggestions if you know better.
The parts, laid in the final case.
A prototype version of the project.
The finished circuit board and connected components.
Back of the case, showing the belt clip.
Front of the case, showing the output jack.