You're all wet. The vast majority of Catholics use birth control in contravention of Church policy. It's the hierarchy that has their panties in a knot, not the rank & file. And Repubs lose a lot on the issue. Look at the furor over Komen's pandering to the Right Wing wrt PP. It's Elian Gonzales & Terri Schiavo all over again.
It's not like any religious organization is forced to run outside businesses, at all. They can divest any time they want. Hospitals only lose money on paper, if at all, and they provide employment for Catholic orders, as well. So the hierarchy has a choice- if they can't take the heat, they can get out of the kitchen. The rank & file won't care either way, but I rather suspect that a majority likes the idea of contraception being a covered service under the new health insurance plan, as do insurors. The Church makes enough money off it all that they'll probably choke it down... particularly with hte set-aside they've already been allowed.
They can handle it, if they don't back themselves into a corner first. Check out the convoluted logic used wrt ectopic pregnancy-
http://johnpaulbioethics.org/FinalProofs.pdf
You can't just remove the embryo, because that would be killing it, but you can remove the fallopian tube or part with the embryo inside, knowing it will die...