I'm not familiar with all of Korean's intricacies of family definition, but would your wife's family all qualify as family from her account? If so, transfer SPG from your account to her SPG account (allowed assuming you have same address on file for >= 30 days) and then she can transfer to her Korean account and book for them.
As for BA denying there's award availability on LAN, it could be phantom availability unfortunately. I've noticed it's often phantom when BA says there are 7 seats online for LAN/TAM flights.