Yes, if Steffie, as an employee of the White House, gave a background briefing that directly contradicted his later statements, the Clinton White House could ethically release the briefing.
When you work for the White House, you speak with the voice of the White House. And the White House controls all the information eminating from it, both background and public.
This is similar to corporations reminding you that whatever work you do for them is their property. Same principle.
Fox said the reason it was "background" is that the White House asked them to do it that way. So today, they asked if they could release it, and they agreed. So it basically was the White House that decides whether it is on the record or background.