Then I tried Reflector, which acts as an AirPlay receiver so I could mirror the iPod to my Mac and record *that*. No dice on new versions of Reflector, but I *DID* manage to get it working on Reflector 1.0 from 2014. But it was wireless mirroring from a 2013 device, so the quality was quite poor.
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/