Finally, at my darkest hour, I gave in. I went to best buy and bought a Lightning to HDMI adapter, a capture card, and an HDMI cable. It was like $130 all in all.
Since the adapter is a tiny AirPlay receiver, I'd STILL be getting compressed video output with MPEG artifacts all over the place. Better than the wireless I got with Reflector, but certainly not a crystal clean output.
@panic has a great post exploring this: https://panic.com/blog/the-lightning-digital-av-adapter-surprise/
I got home, plugged the adapter into my iPod, then the HDMI into my TV to test. Voilà. In all its crunchy glory. I unplug and replug the adapter, and...nothing. Black screen.
I checked every tweak installed, tried rebooting into safe mode, I could not get it to function again. I tried plugging my iPhone X into the adapter and that worked fine, so what gives??
Eventually I tried restoring the iPod back to the same firmware, and it worked again. Hmm...