You won't likely get any help. This is a sub for Android TV boxes, not boxes that run Android.

Since you're running basic android, surely there's just an app in Google Play...

ADB has built-in screen recording tools. You can look at doing it from the command line.

Ask /r/settopboxes or /r/AndroidSetTopBox

You don't have an "ATV" box, you have an Android "TV box". We only talk about official Android TV devices in here.