Partially figured it out by inspecting the images in the media browser. The image source in the Media Browser UI is something like…
/media/local/image_name.jpg?authSig=eyJ0eXAiOiJKV1QiLCJhaGciOiJIUzI1NiJ9.eyJpc3MiOiJlZjA5ZDJmNjMyZDk0MjdjYmYwMmRiMDZmZWE4NWQ1NiIsInBhdGgiOiIveWVkaWEwbG9jYWwvQmVuLmpwZyIsImlhdCI6MTYwMTE3NTc4NSwiZXhwIjoxNjAxMTc1ODEefQ.A5ScY2t6nJMOuwT1M3B59r2YdVf14AP2GE-5-k0JHl4
Dropping that into the Image Path of the Picture Entity card works but seems cumbersome and not the right way to go about it. Also makes the YAML editing in the UI impossible because it makes the UI edit element stretch way out to accommodate the long string and pushes Cancel, Save, and other elements off the screen.
Edit: Looks like that authSig value in the image string is user specific. If I try to view that Picture card from my wall panel (which is logged in as a different user) it doesn’t work. Guessing I’m going outside the bounds of what Media Source is meant for.