While it would be unconstitutional to tax churches but not to tax other non-profits, it would be fully constitutional to tax all non-profits equally.
What about the McCulloch v. Maryland precedent?
The law that was struck down did not apply to all banks equally. Banks not chartered in Maryland but operating in Maryland were taxed $15,000. At the time, the only such bank operating in Maryland was the Bank of the United States. If the Maryland law applied to all banks operating in Maryland equally, then I doubt the law would have been struck down.
Huh? I don't think a lack of equality in the taxation of banks had anything to do with the Court's ruling in McCulloch. That case was decided almost fifty years before ratification of the Fourteenth Amendment. The Court invoked the Supremacy Clause of Article VI and said that a state cannot interfere with the operation of the federal government. I've always thought to myself that the gist of McCulloch is that the taxpayers of the United States never owe anything to the tax collectors of a state, because the federal government and its valid, constitutional laws are supreme. (I've also thought to myself that "The power to tax is the power to destroy" is drama queen rhetoric, unbecoming of a staid Supreme Court opinion. There would have been an easier way to convey the sense that imposing a tax on an entity is interfering with that entity.)
McCulloch explicitly states:
The States have no power, by taxation or otherwise, to retard, impede, burthen, or in any manner control the operations of the constitutional laws enacted by Congress to carry into effect the powers vested in the national Government.
This principle does not extend to a tax paid by the real property of the Bank of the United States in common with the other real property in a particular state, nor to a tax imposed on the proprietary interest which the citizens of that State may hold in this institution, in common with other property of the same description throughout the State.
So ordinary taxes are allowed, but not a tax aimed specifically at impeding an agency of the United States.