OK, I'll try that this evening. The vehicle is a 2006 Toyota Tacoma. I thought about grounding to the body initially, but the ground wire was right there next to the power wire. I'll update after I test it out.
If all else fails, I may just plug the video wire to the AV input plug instead of the Rear View Camera plug and wire it to always-on power somewhere. I tow a little popup camper so it might be nice to see behind me whenever I want.
Thanks, and anything else you can think of will be appreciated.