Instead, put the tape recorder on the table and announce you're going to tape the interview. If they ask why, tell them about the vast amount of "misplaced" paperwork wrt job performance from the past couple of years.
They can't tell you not to tape it, AFAIK; at least, they certainly couldn't up here in Canada. Mind you, you wouldn't need to ask their permission or inform them you were taping up here either; only one party to a conversation need give permission/be aware of a taping of a conversation.