BCBS did right. Anata is right also but the provider billed to either of them wrong. At the same time, your husband should fill one form to show you have another insurance and this is secondary.
Anata rejected your bills mostly because your coverage is different.
Be patient and call Anata and ask them: 1) Why they think they are not secondary; 2) How will they pay if the doctor billed wrong and what doctor need to change, mostly the diagnosis, procedure, drug codes; 3) Call the billing department of the provider and a)tell them you have two insurance and ask them how to bill correctly for both of them; b) ask them what did they billed and write down all diagnosis and procedure codes; c) ask provider billing department if Anata is primary, what codes should they bill and what codes should they bill to BCBS;
Call Anata and ask to bring BCBS join the conference call to talk with each other, even the provider billing department.

All your bill face same situation: you assume BCBS is primary, and providers billed to BCBSA and they rejected. You need to fix this first.

Insurance company 100% follow the billing codes to pay and they have no right to change the claim codes from providers.

good luck.