I did my little prototyping, and it worked like a charm.
The Sony AVX AV3250 has a parking break sensor, that enables the display to show the videosignal if, and only if, the parking break is enabled. Or, if you do like I did, and just ground it to the chassis of the car.
The next bit is to plan out the permanent power connection, and design the panel and bracket that will hold the connections and devices. I’ve been screwing around in TinkerCad, and I ordered some “powerthiefs”, essentially a power splitter with a separate fuse.