0bastard is deliberately focusing on the Churches as a subterfuge to obscure that fact. Firstly he knows he can tap into prejudices against Christianity, especially against the RCC, and secondly he knows the RCC is the biggest organization that will object to this so they become his Alinsky target to freeze and polarize.
Nothing in the 1st Amendment says I have to belong to a government-recognized religious organization in order to qualify for 1st Amendment protections. In fact what it does say is that I don't! But 0bastard is laying down the precedent here that there are government-created standards I must meet to be covered by it.
Once that is accomplished what else can they use to separate a citizen from a BoR's protection? What other actions will they force me to commit or refrain from once I have been deemed unqualified for a given BoR's protection? Will I have to belong to a government approved gun club to own and shoot a firearm? To publish my opinions on the internet? To stand with a group of protesters or sign a petition of grievances about the government?
He's pushing the issue of contraceptives because he wants to make the Catholic Church look 'anti woman'. That's how he thinks he'll win this argument in today's media saturated culture.