There is a simple compromise on this story that should make all sides perfectly happy: just have the government provide birth control for free to all women who want it. Catholic colleges won’t have to buy it for women who work for them, and women would be guaranteed access. If everyone here is being completely honest about their motivations, then win-win.
It’d also guarantee a larger market for pharmaceutical companies and take the burden of birth control off employers so they would theoretically have more cash on hand to hire more workers, so make that win-win-win-win. Plus a large single payer could drive the price down in ways that individual taxpaying women and insurers couldn’t, so add another win up there.
If the church still says that they’re funding birth control through taxes, well, they’re also paying for capital punishment through taxes and they haven’t closed down all their schools and hospitals in the US because of that one.