Imagine if the state somehow managed to justify having church-goers register with the state. Would that justification automatically justify making the records public? I don't think so.
Here in Kalifornia the records of concealed carry holders were being kept secret and this facilitated the abuse of the system by allowing sheriffs to grant permits to their donors and deny others.
Since keeping the records secret encourages corruption and keeping the records open invades privacy, it really shouldn't take the courts very long to decide that keeping the records at all is an infringement of the right.
The same people who feel totally free to publish and subject to harassment or worse gun permit holders (because they don’t like their position on the subject) fight like hell to assure that no one can know who’s had an abortion or who has AIDS.