How do I know if a PS Vita game is compatible with the PS TV?

If you have the case, US PS Vita games that came out after the PS TV's release will usually have a "Compatible with PS TV" label in the top left. For older games, you won't know until you try them.

Here is a UK-list and a US-list. I'm linking to the pages via Archive.org so you may also want to check the actual live pages below for any additional updates, although they are highly unlikely at this point.

https://www.vitaplayer.co.uk/list-of-playstation-tv-compatible-games/

https://www.pushsquare.com/forums/ps_vita/a_master_list_of_games_that_are_compatible_with_pstv