Trouble is, the FA Cup TV contract is a much cheaper and less comprehensive deal than the Premier League, so if our game isn't properly covered by British TV then no international broadcaster can show it.
Whichever British TV company holds the Premier League highlights rights (currently BBC and Sky) is contractually obliged to film every single game - even the 3pm Saturday games not shown live on British TV - to the standards expected of a modern live TV broadcast (i.e. at least 10 cameras, a full on-location Outside Broadcast production team to insert replays, captions etc as they happen, and an on-location commentary team). Because all of this is done live at the stadium, the Premier League can sell every game to international TV companies.
Meanwhile, the FA Cup contract only requires the British TV highlights rights holder (currently ITV) to film 3 or 4 additional games per round for international audiences (on top of the 5 already shown live on British TV) to the same standards described above (remember, there are 32 games played in the 3rd round so it would be far too expensive to send a full TV crew to all of them. By the 5th Round of the cup there are only 8 games to be played, so every game can be covered by a full TV crew).
Every FA Cup game not picked for comprehensive coverage by the British TV companies will only have barebones coverage for highlights purposes: a few cameras at the ground, no on-site commentator, and no Outside Broadcast team to add live replays etc. Obviously LFC TV can add a commentator and edit in replays etc after the match for their delayed broadcast of the game at midnight UK time, but I doubt they have the rights to put a live package together to show overseas. Even though everyone on this forum would argue that any Liverpool game should be important enough to be selected for extensive TV coverage, the British TV companies are primarily looking for potential shocks, and a Premier League team at home to a lower division side "should" be a easy-but-boring-for-neutral-viewers home win (famous last words, I know!).