Yes, with a but...

Do everything as usual until you get on board the aircraft. Politely let the purser know that you will be letting your wife take your BizE seat and you will be taking her seat in coach. I've never heard of an FA arguing you cannot do this. You may even be complimented. I've had Fa's offer me free drinks in coach while my GF sits in my first seat, but it's a gift and not to be expected.

Don't worry about the TSA. They have nothing to do with it.