If your religion says that you shouldn't promote contraception, stick with your religion. That said, you have the right to because you have religious freedom. So, you shouldn't be forced. Is someone wants contraception, they can go to another place. You don't have to give them a condom if you don't want to.
Separation of Church and State works BOTH ways. Under the Lemon Test, set forth by The Supreme Court of The United States of America in 1971, The government's action must have a secular legislative purpose; The government's action must not have the primary effect of either advancing or inhibiting religion; The government's action must not result in an "excessive government entanglement" with religion.
The latter 2 are violated. It is unconstitutional. In addition to Congress not being allowed to make a law that trespasses upon the rights of the Church.
The freedom of choice in the United States of America is what makes it a great country. Religious nonprofits should be able to choose exactly what kind of health care coverage they wish to offer. As such, those who work for these nonprofits are free to leave their job and go elsewhere if they do not like the options given to them.
What kind of health coverage your employee gets shouldn't be dependent upon your personal morals. If you don't like contraception, then you don't have to use it.
However, just because you are morally opposed to it does not give you the right to take away that choice from somebody else. When you deny your employee the right to health coverage that doesn't include contraception, you deny them a choice.
Forcing religious nonprofits to provide that kind of coverage is not about taking away their religious rights; it's about making sure that they can't force their religious beliefs and morals on others.
I do not think that religious nonprofits should be allowed to offer health coverage that does not include contraception. Birth control is legal and health care is audited by the government so I believe that birth control availability should not be optional. Whether the individual takes advantage of it should be their choice.