I'm trying to have a class feature to let the player choose an additional Metamagic option to learn (from the list of original Metamagics).

I don't see a way to do that so my work around is to manually add each Metamagic in as General action option.

If there's a better way to do this please help.

You can't add homebrew metamagic (or fighting styles, or eldritch invocations) yet.

You can create an option for each homebrew metamagic and have players choose between original and homebrew options, but not exceding their choice limit. They will have to follow the written instructions rather than doing as much as the character sheet will let them (which players are bad about).