If your flight from BIO was on IB I don't see how BA would help at that stage with checking bags?

If IB at BIO say you can't overnight check a bag for MAD then that's their call. It does seem there is overnight bag store at MAD based on some searching on the IB board, but I don't know if that has changed since or could be suspended atm. Was the connection time at MAD over 12 hours? That does seem to be a limit.

https://www.flyertalk.com/forum/iber...d-layover.html