Electronics are weird. I did the whole process again but I did not power cycle the console. But, now my APEX works with the latest controller firmware!
1. Connect controller (without chatpad) directly to console via usb.
2. Downgrade firmware on controller.
3. Restart console via guide hold context menu.
* Previously, I had skipped this step.
4. Upgrade firmware on controller.
5. Restart console via guide hold context menu.
* Previously, at this point I had selected "Turn off console", then disconnected console power for 30 seconds after console was off. I only restarted the console once, before, also.
6. Verify controller still works, controller still connected directly to console via usb.
7. Plug controller into APEX USB Hub.
8. Connect APEX to Xbox.
9. Press guide button on controller, verify 4x green lights on APEX.
10. Profit! APEX works longer than three seconds.
Thanks for the help, y'all!