I actually looked into this about 5 years ago... in Budapest.... Generally speaking people thought i was nuts. The Hungarians are almost Germanic in their logic and warned me that i could not get a license in Hungary with out a lot of trouble; and couldnt understand why do it without getting a license.
Anyway, for the sort of thing you are talking about (and I was talking about) you simply need a person to perform the symbolic ceremony of your choice. That generally means a cleric willing to do it for a few bucks. Off the top of my head I can recommend a good Rabbi. I could be wrong, but i dont think he will care about your religion and we share a few of the same books...... Then you could say you were married by the Rabbi that was on Oprah.
If you are serious I could make a few calls and maybe come up with the name of a Christian cleric or two for you to contact. Let me think of the location too. What time of year? The boat docks are sort of heavy with the tourists and maybe we can find something tucked away a bit but with the view and ambiance. Personally i would go for the back garden of the Great Synagogue or this shabby little shul i know of in the 8th district --- but there i go with the Rabbi again. There is also a lovely little church on the embankment in Buda...
By the way, i am a baptist...