Religion is not special
A spiritual practice is something that an individual or community voluntarily engages in in order to bring themselves closer to (a) God (or The Universe, or Nature, or The Ultimate Sources of Existence, or whatever your spirituality is directed toward). A religious law is an attempt to compel society to get closer to God (etc. -- I'll just say God from here on out since the controversy at hand involves a monotheistic religion). As a general rule, I think we ought to have a presumption for limiting burdens placed upon sincere spiritual practices, but ought to grant no special consideration or exemptions to religious laws.
There are two key distinguishing features of a spiritual practice -- its core function is focused on the practicing individual, and it does not directly affect others. There is certainly some fuzziness to this as with all categories. After all, your spiritual practice may affect me simply by being really weird and thus making me creeped out that you're doing it. Shamanistic urine-drinking, for example, would make most people in the US (including myself) a bit grossed out. But the value to practitioners of shamanism is so much greater (since it's an important practice in their spirituality) that those of us grossed out by it have no leg to stand on to oppose it as public policy -- and indeed, there would be grounds for a shamanic exception to some general public health law that might forbid urine drinking. Note that this is essentially the same logic that we would follow in the case of people who practice urine play in a consensual kink/BDSM context -- it's gross to think about for most people, but that's clearly outweighed by the satisfaction derived from it by people with a golden shower fetish. The fact that a practice has a spiritual dimension doesn't intrinsically make it special, it just serves as a marker that said practice is very important to the practitioner.
Addressing the contraception issue in terms of conscience clauses framed it as if it was a spiritual practice. Catholic institutions (or rather, their members) wanted to get closer to God by practicing not-offering-birth-control. But this alleged spiritual practice fails on both counts. First, it's not focused on the spiritual practitioner. The spiritual value of not-offering-birth-control is entirely derivative of the value of not-using-birth-control -- it's bad to hand it out only because that would facilitate it being used. And second, of course, it harms someone else in a pretty significant way, depriving owners of uteruses of an important medical treatment that they would otherwise be entitled to.
The compromise offered by Obama sought to deal with the issue as a matter of a spiritual practice, offering a way that the public policy goal (contraceptive access) could be achieved without burdening Catholic institutions' spiritual practice of not-offering-it. The rejection of this compromise by Republicans shows that this was never about a spiritual practice in the first place. It's not enough that certain people can avoid direct involvement in contraceptive provision -- the policy needs to stop contraceptives from being provided. This is why I give Republicans and conservative Catholics a bit of credit for honesty here. Their real concern was to implement their anti-contraception agenda, and so they refused to go along with the charade of a policy that would allow them to keep their personal hands clean while the effects of the policy got implemented anyway.
We can thus see that we're dealing with a religious law, one which says contraception is bad and should be restricted. The arguments against contraception are religious law type arguments -- that it is bad for society to allow people to alter the God-given design of sex, for example. The problem then is that religious law cannot be the grounds for policy exemptions. Religious law is an argument about what the overall policy should be. And as such, religious law has to go up against all of the other (secular or religious) arguments for or against the policy in question. Based on those arguments, society then makes a collective choice as to what the policy will be for everyone. The losers of a policy debate don't get to be exempt from it.
There's a common liberal (Rawls-inspired) claim that religious laws are invalid arguments to be put forward in the public sphere to argue for or against policies. I think this partakes of the same religious exceptionalism as the claim that religious laws should get policy exemptions. If you put something forward as a law, then you are making the claim that it can be backed up by reasons and evidence and be persuasive to others. Most religious people do in fact believe that about their claims. In this sense religious laws are no different than secular justifications for policy (which contain their own unsupported factual and value claims). Wanting to be closer to (a certain version of) God is not a prima facie more admissible goal than wanting to bring the greatest happiness to the greatest number. But we must then proceed to analyze the pro and con arguments, and in doing so it should be apparent (though I don't have room in this post to go into it) why access to contraceptives should be mandated.
So the choice is this: either you have a spiritual practice that you can do as an individual or consenting community but cannot burden others with, or you have a religious law that could be adopted or rejected as policy for everyone. You don't get to claim exemptions and special circumstances in which to implement your religious law when you can't make it the policy for everyone.