I'd rule against Bradwell. The Priviledges and Immunities should only apply if she was still a citizen of Vermont. Since she was living in Illinois at the time, Illinois law should be the only one that applies.
However, if one were to read the wikipedia version, me agreeing that it was sound would make me sound like a sexist because of Justice Bradley's statements.
Just FYI, that's not what the Privileges and Immunities Clause of the 14th amendment does, regardless of whether you believe Slaughterhouse was the correct decision.