You both have a fundamental misunderstanding of what "free exercise of religion" means. No part of the law prevents Catholics from practicing their religion. Their religion says, "Don't take birth control" (or so they believe; the Bible is surprisingly silent on birth control pills). They are still free to not take birth control, and they are still free to tell people not to take it (even though 95% of their women do), therefore they can still practice their religion.
If we used your model of the 1st Amendment, the church could say, "It's against our religion to have fire exits, because fires are a part of God's plan," and the state would not be allowed to make them have fire exits in their churches.
If Congress said, "All women must take the pill," that would be a 1st Amendment issue. That is not what's happening.
You are correct in saying that the HHS Mandate isn't forcing people to take birth control pills or contraceptives against their will. However, this is anon sequitur because the Catholic Church and those against the mandate have
never argued that this is what the HHS mandate will do. What they argued is that by forcing Catholic based insurance companies and Catholic schools, hospitals, and charities who are self-insured to pay for and allow access to artificial birth control, contraceptives, and abortifacients, they are forcing them to provide the means and opportunity to potentially commit what the Church believes to be a grave and mortal sin. And, according to the Catholic Church, if you provide the ways and means to allow someone to commit what you believe is a grave and mortal sin--even if that person doesn't actually go through with it--then you are an accessory and are just as guilty of committing a grave and mortal sin.
Second, you're also correct in saying there's no mention of birth control pills in the Bible. There's also no mention of human cloning, in-virto fertilization, stem cell research, evolution, the possibility of life on other planets, global warming, Nuclear disarmament, etc. But the Catholic Church have still weighed in on these issues because they believe that that Bible isn't the sole authority on forming Church doctrine.
Likewise, citing the fact that 95% of Catholic Women use some form of birth control doesn't mean that the Catholic Church is just going to placate on this, especially if they sincerely believe that using artificial contraceptives and abortifacients is wrong. Also, if that many women are practicing birth control, doesn't that automatically suggest it's already widely available and affordable without the help of the federal government or the HHS mandate?
Finally, the fire escape analogy is a false equivalency. Aside from the fact that no religion, as far as I know, has ever stated such a thing, providing a service, like a fire escape, wouldn't be seen as wrong according the Catholic Church because it's being used to save lives and is in perfect compliance with a law that doesn't violate the tenants of their beliefs. Proving the means and access to birth control is wrong, according to the Catholic Church, because it potentially leads people astray to violate what the Church teaches.
Also this.
If you have such a problem with this, maybe you should take it up with the 95% of Catholic women who want birth control?
Yes, the law states that insurance companies must provide for those services, but there are some insurance companies who are exclusively Catholic in their practices and there are Catholic schools, hospitals and charities, etc. that are
self-insured. Meaning that the Catholic Church would still be indirectly paying for artificial contraception and birth control under the mandate.