Apple TV 3 Can't Handle Two-Factor Authentication?

I use my old Apple TV 3 only to Airplay music to my sound system since the ATV 3 still has an optical output. From there to my excellent DAC to my amplifier. Works great.

I recently established two-factor authentication for my Apple ID because someone was apparently trying to hack it. Wasn't too hard to do, and seems to have fixed the problem. But, there doesn't seem to be a way to log in to the Apple TV 3 using two-factor authentication. Now when I activate the Apple TV 3 to stream music, somehow Apple thinks someone is trying to log in and sends me a report on my computer, along with asking for permission, and the two-factor identification code. The ATV 3 streams anyway, so all works as before, but receiving this report and asking for permission each time is annoying.

Does anyone know a way around this?