The Apple TV app on smart TVs and third party streaming devices would support two-factor authentication through in-app options. It does not depend on LG functionality.
If you get an “unable to verify” error message, with or without prompt for TFA verification code, then the error may not be about TFA, but instead about communication with the remote Apple servers.
- Please verify that the TV system software is up-to-date.
- Verify that the Apple TV app is up-to-date, if not done automatically.
- Verify that your router or firewall doesn’t block Apple authentication servers, in case you made custom changes.
- Verify that your devices and router use a good DNS, like the one from your ISP, which gets updated with all the server names that Apple might use. Again, only applies if you made custom changes.
- Verify that your TV has the correct date, time, and time zone, as Apple may be more strict about that than other content providers (for TLS secure connection reasons).
Else I would suggest to contact an expert from the Apple Support staff for further assistance. Contact Apple for support and service (phone numbers) or start your support request online (chat/email/callback).