Simply, the rules are the member must be staying at the same property (in their own room) as the other two rooms and settle the bill. You are not allowed to earn credit for booking rooms for other people, or yourself, at other properties (although some have found ways around that) for the same night regardless how close/far they are to each other.

Unfortunately, it seems a CSR was making up rules when they said the rate made the stay ineligible. Happens all too often and they should be called on that as being unacceptable. But if they feel they can get you off the phone, then they will.