a trip to Best Buy showed me there's no adapter from Coax to Composite (there's vice versa, but it's not backwards compatible).
Sure there is. The easiest way would be to find a VCR. What you need to find a video modulator. NOT an RF modulator. image.jpg
The top one in the pic is just an rf modulator, it will down convert video to coaxial. These are common
The bottom one here is a multi-switcher that can take a coax signal and output it on the rca wires. Might be hard to find now a days. I've had this one for ~15 years.
Any VCR can do this for you too.