For those wishing to follow along:
.pdf file of schematicYes, your reverse light layout is fairly straight forward, the reverse lamp being powered by a single (+) DC wire and ground. We don't know how that is switched or protected within the chassis but that's pretty much immaterial. It's been awhile since I did any rear lighting on Japanese vehicles but I'd suggest you consider another source for powering the camera and monitor. IIRC, using parts of the rear lighting (even for simple trailer lighting) was always a pain.
It's simple enough to grab either + ign. or constant with a fuse from somewhere else.
Have you tried powering the camera and monitor straight from a battery or some other 12V source?