You seem to have a bug up your ass about Republicans, but I don't understand it. Most Republicans in DC are indistinguishable from most Democrats, once you look past the theater to what bills actually get passed, which are whatever the billionaire donors want. Everything else is just theater, on both sides.
Not all Whig politicians were able to get re-elected as Republicans, BTW. The platforms weren't the same and some had doubled-down on increasingly unpopular ideas (otherwise, the party never would have faded). I can't predict what coalition will arise from the ashes of the GOP, but Trump proves that catering to the religious whackos has become unnecessary and pointless - it never actually mattered that Trump is pro-choice, and not particularly religious.
And yes, today's GOP is "dead party walking", unless Trump somehow wins (Hillary would have to stroke out) and even then only the name would survive. Trump supporters are furious with the GOP, and without them it's a 40% party.
And of course, there's the legal cases still working their way through the courts where various entities are arguing that even saying to the insurer (who has no practical objection to birth control, it's a cost-saver for them), that they don't want to be involved, is a burden on them, to fill out a form, saying leave us out of it.
The court cases are about filling out a government form registering your religious beliefs. I object to that too: history suggests that sort of thing never ends well.
I'll at least expect a conscientious objector to report their status to the draft board.
Different case. The rule is that the State cannot not compel you to act against your strong moral beliefs unless there's a compelling state interest and the action is the narrowest possibly to address that. Registering as a conscientious objector is a perfect example where both are true. The court found no compelling State interest in having birth control paid for by insurance (rather than, you know, money).
Be open at some hours. Be closed at others. Access for people, even service animals.
You'd be surprised by what gets waved for legitimate religious objections. A strictly kosher restaurant, for example, doesn't follow all the same rules (of course, it has rather more self-imposed).
As for birth control, if it's against your moral principles for a person under your employ to make their own choices about their reproduction
Now you're talking about a very narrow subset of Catholics, and we're effectively back to fringe cults. But if you had it as corporate policy that employees couldn't use birth control, that would be very different legally from not paying for it via insurance. The former is an undue burden on the employee, the latter isn't.
Personally, I'm against any law mandating insurance coverage in all policies for that only women in a certain age range need - that's singling out a group of privileged people for elevated legal treatment, and again history shows that sort of thing never ends well.