How can I watch BBC with appletv

Nothing else offers the full catalog of shows except BBC iplayer.

if you aren’t in the UK and paying the fee, you aren’t legally allowed to access it.

Outside the UK in limited number of countries, there is BritBox. It is a subscription service that allows access to some of the BBC content.

The player is available for UK accounts only i.e. on the Apple UK store. Where are you located? Without an Apple UK account, the real thing is impossible, only what BBC chooses to offer abroad.

If in the US, see also https://www.bbcamerica.com/

See https://www.bbcselect.com/world/ - available as a paid option on Amazon Prime.