Mississippi was able to defeat a personhood amendment by generalizing the argument - pointing out the effects on contraception, IVF, stem cell research, property rights, etc. They made personhood undesirable to people who support abortion restrictions. We need to similarly generalize why people who support the rights of the Catholic Church to reject birth control should nonetheless reject an exemption to the ACA. Follow me below the squiggle for more.
Every single aspect of modern medical care can be rejected based on the teachings of one religious group or another. Most people know that Jehovah's Witnesses refuse blood transfusions. But there are also Muslims and Orthodox Jews who reject pig heart valves, and pork derived insulin or thyroid medication. Most tablets and capsules involve gel coatings or shells that are derived from pork or beef byproducts that can be objected to on religious grounds. Christian Scientists and some Pentacostals and other groups reject even chemotherapy and antibiotics.
If we say that an employer can exclude contraception coverage from their employees' health plan, it opens the door to the employer rejecting other types of coverage as well. And they have a financial incentive to do so. A health insurance plan that covers nothing but prayer, if an employer can rationize it on faith grounds, is a real money saver.
If we say that an employer can exclude coverage for anything they want, it is no longer possible to justify employer-based health insurance at all. That's an argument for single payer, government insurance.