Also COC/T&Cs are only applicable to your contract with BA, therefore there’s no discussion possible about the final destination.
BA didn’t incorrectly use Timatic, putting MRU as destination in Timatic was a shortcut you shouldn’t have taken without consulting BA!
Airlines are not legally obliged to take onward travel, on different contracts, into consideration. They often do, but in your case there were, unfortunately, too many red flags involved. Hopefully BA gets your family on the direct MRU, a commercial gesture I would consider as all’s well that ends well.