Would you say that "bid goodbye" is an example of linguistic register mixing?
linguistic register mixing
I don't know what you mean. I have never ever heard anybody "bid goodbye" to someone. You do hear "I bid you farewell" occasionally in period dramas. Maybe in Romantic fiction?. I don't if it is an example of linguistic register mixing but it sounds pretty mixed up to me.
"Googling "bid you good-bye/goodbye" throws up Anne Brontė, Henry James, a translation of Turgenev, lyrics by Lou Reed/The Velvet Underground and by Super Furry Animals."