Amelia Thomson-Deveaux has a great piece about religious groups that are trying to remove restrictions on church-based electioneering. She suggests that rather than gutting the rules, there's a simple fix, "Religious leaders who want the liberty to endorse candidates can give up their churches’ tax deduction."
I would go one further. Let's tax churches! All of them, in a non-discriminatory way that doesn't consider faith or creed or level of political engagement. There's simply no good reason to be giving large tax subsidies to the Church of Scientology or the Diocese of San Diego or Temple Rodef Shalom in Virginia or the John Wesley African Methodist Episcopal Zion church around the corner from me. Whichever faith you think is the one true faith, it's undeniable that the majority of this church-spending is going to support false doctrines. Under the circumstances, tax subsidies for religion are highly inefficient.
What's more, even insofar as tax subsidies do target the true faith they're still a pretty bad idea. The basic problem with subsidized religion is that there's no reason to believe that religion-related expenditures enhance productivity. When a factory spends more money on plant and equipment then it can produce more goods per worker. But soul-saving doesn't really work this way. Upgrading a church's physical plant doesn't enhance the soul-saving capacity of its clergy. You just get a nicer building or a grander Christmas pageant. There's nothing wrong with that. When I was young I always enjoyed the Grace Church Christmas pageant. But this is just a kind of private entertainment (comparable to spending money on snacks for your book club—and indeed what are Bible study groups but the original book clubs?) that doesn't need an implicit subsidiy.
Meanwhile, nobody thinks churches and other religious institutions should silence themselves on the important issues of the day. On the contrary, discussing moral action is at the heart of many religious enterprises. And much moral action plays itself out in the arena of politics. So trying to say that churches should get subsidy when they don't endorse candidates is de facto a kind of subsidy to religious doctrines whose views happen to lack strong partisan implications. So if your faith says "abortion should be illegal and spending on the poor should be increased and it's too bad neither candidate supports that" you're golden, but if your faith says "abortion should be legal and spending on the poor should be increased so good for Barack Obama" suddenly you're in trouble. That's perverse. Just make everyone pay taxes.