When you rang did you ask in parallel if they could manually assign the Partner Card to someone outside of your HHA/F&F? I can see why the website might automatically offer you a choice from the list, but do the T&Cs of the Partner Card really restrict you to those members.

Alternatively would there be a big downside in disbanding your HHA, for the minimum 6 months. If you have the plenty of avios in your own name (or another member's name) to make any redemptions you intend in that period, disbanding the HHA would free your hand in regards to the Partner Card. (Potentially complicated if you have under 18s in your HHA.)