I was unable to reach anyone capable of doing this, BA call centers were crazy and I didn’t want to lose availability.

that said, I believe this exists, because BA does it for Amex for British cardholders. It is reputed to be a very bad rate of redemption, but I can’t report on US rates.

Instead, I booked me, P2 and both kids each on a separate booking, so that we could get $100 reimbursement per ticket. It’s on British metal direct to LHR one way, taxes etc were $149 before reimbursement, so only $49 if reimbursement comes through. It’s a late June off peak redemption, so only 13k one way. (We’ll book United back from Italy I think.)