The problem is that it's not the government forcing its way into the Church.
It is the Church inviting the government in.
I say this because any contract, either between the government and a company, or between two business entities, has terms and conditions associated with that contract.
In the case of federal contracts, those terms and conditions are outlined in the Federal Acquisition Regulation.
In this particular case, the terms and conditions are listed in FAR Part 22.
The reason why it's not government meddling is that nobody forces the Church to sign that contract.
(NB: Please Note: I am not saying the above to defend the government (I'm not). Imposing this is plain evil. But it shouldn't be imposed on anybody, not just the Church. If the Church, knowing that this FAR clause exists, signs contracts to perform services with the government, particularly as long as this EO exists, then, in truth, it's the Church's fault)
I was opposed to the “Faith based initiative” for exactly this reason.