In fact, having re-read Curits, I think I know what Bingham was getting at, and your guy is completely missing the point. Bingham believed that the states were always bound to respect the Bill of Rights, as well other freedoms not listed there explicitly, but that prior to the 14th Amendment there was no legal way to enforce this. Thus the states were bound in theory not in practice. In this light, his quote make perfects sense. Of course, I'd have to read the whole speach in context to verify this, but your link doesn't help in this regard.
Of course, it may also be that Bingham was something of an "eccentric" in his views. If you're correct that he believed the Bill of Rights had ALWAYS applied to the states (which it absolutely didn't) then it's not surprising he would claim the same thing after the 14th was adopted.
Not a single level of government seems to have agreed with him. Are you aware of any legislation from that era related to this issue? Was there any effort by Congress to exercise its clear 14th Amendment enforcement power by passing a religious establishment bill of some sort?