I was thinking this: mike&co can have his account open in one window and his wife's open in another. Using either account, he can find a flight that has four award seats. He can book two seats using his voucher in one window, and simultaneously book two more seats using his wife's voucher in the other window. Both his account and his wife's account can each access the total number of Avios in the accounts that are linked together in the household account, so there is no danger of being short of Avios in one window but not in the other.

This answers the question "how can I access and redeem the second voucher?" with the minimum risk of the booking falling between two stool.s

Of course, this leads to two bookings, one in his name and one in his wife's name. If that is disadvantageous for some reason, and all four passengers must be booked on the same booking, then I think you are right that it would have to be done over the phone, if it can be done at all.