Science..., you wrote, "...feel threatened to put the name, and fact of abortion, in the public record..."  That is a valid concern.  

Seven years ago, I wrote this: http://supreme-court-gender-equality-pac.blogspot.com/  In seeking advice, I talked with a wise old woman who was retired from a distinguished law career in Washington, DC. Without reading my proposal, she lectured me on how unintended consequences inevitably plague all new laws. So I know: I can't control outcomes after my ideas enter the political arena.

If it were mine to do, I would establish the public record by state.  I would list only names and abortion dates. I would NOT list addresses or contact information, not city, not county >> just state, as in Oregon.  Because of that, no newspaper would include the record in publication.

Also, I would plainly state: "This record includes all abortions, including those done to save the mother's life." No one would be able to know the "why" of any listing on the record.