We currently give a loan customer a new Privacy Notice with every new loan even if we do two loans in the same day. Is this overkill or required? We have a difference of opinion here so any guidance would be appreciated.

I think it is. According to REG P you are required to give the notice;
(c) When you establish a customer relationship. (1) General rule. You establish a customer relationship when you and the consumer enter into a continuing relationship.

So to me if you already have a loan, I interpret this to mean that you are not required to provide another one.

There is an exception for existing customers in 12 CFR 1016.4(d)(2) that says:

"If the initial, revised, or annual notice that you most recently provided to that customer was accurate with respect to the new financial product or service, you do not need to provide a new privacy notice under paragraph (a) of this section."