You're saying the safety car only works for the F12020 mod and any other series you drive in the game still use the original safety car?
Then I would like to know how that is done because as far as I know GTR2 can only have 1 safety car and it will use it for every session you drive, regardless of wich series and car you drive.
So to answer your question, if the mod you downloaded and installed uses a custom safety car it should automatically be used across the whole game.