It gets a bit stickier when said religious organization must deal with people of different beliefs. If you accept Medicare / Medicaid (which the Sisters undoubtedly do) then you have to treat all of those people without respect to your religious belief (assuming that treatment is considered standard of care). If said patient wants / needs contraception then you must make arrangements for the person to get it. You don't have to prescribe the pills yourself, if that compromises your belief, but you may have to send them down to the (secular) doctor down the street who doesn't have an issue.
Clinicians who work for clinics or hospitals associated with religious orders have long worked around these 'issues'. At the Catholic hospital where I worked, we hid the oral contraceptives in a separate closet that we made sure was closed before one of the nuns walked in. And they would not walk in unannounced. Abortions, obviously, were not allowed on campus, but we could refer people to other providers.
The mandate that birth control be provided really is a straw man. Religious orders have been dealing with this for decades. I don't see how this mandate is functionally any different from, for example, a dominant private insurer who offers contraception as part of their insurance packages. All of the hospitals in town realistically have to deal with the insurer and accept their conditions (we're not going to discuss the implications of that right now - it is a very common situation in the US). You do your dance, as above. You get your money. No money, no mission.
Personally, I think the ACA blew it with the requirement that every insurance policy cover contraception (and maternity benefits). The argument for this has been that you need to expand the coverage base in order to keep insurance more affordable. However, the patient base of persons needing or desiring contraception is quite high enough to allow for economies of scale. 30% of the population (approximate number pulled out of my nether region) is big enough to fund a benefit.
Further, the ACA 'isn't' a tax (except it walks like a tax, looks like a tax and squawks like a tax). There is a longstanding precedent for being taxed for something you might not need personally but is considered a societal benefit (think school taxes). Again the construction of the ACA is that of a horribly flawed kludge (that's the nice word) that benefits the status quo in general and the insurance companies in particular. Rationale arguments get buried in the miasma of details that comprise the legislation and give everybody something to hate. Unfortunately, it was probably the best compromise Obama could make. Whether or not it actually improves health care for a majority of Americans is quite unclear.
