The police can use the false ID against you in court. From what you have said, it is the basis of the obstruction charge.

The police did not have to read your girlfriend her Miranda rights until she was in custody, and even then only if they wanted to interrogate her while she was in custody. And even if there were a violation of her Miranda rights, that would have little to no impact on you.

I suggest you obtain legal help, either hiring a lawyer or petitioning for an appointed lawyer.