With the inspiration that your inquiry initiated, I have found a new source for the possible marriage of James and Catherine. The state of New Jersey
now has an online index to mid 1800's marriage registrations. I found an index entry for J Billingham & C Bradley, married 17 Jul 1852 in Chatham,
Morris Co, New Jersey. This location is reasonably close to Newark. I have ordered the FHS microfilms from Salt Lake for Morris Co marriages. I'll let
you know when I see them if there is any clarifying data.
The films have arrived and your observation was correct, J Billingham and C Bradley were married by B J McQuaid, a priest in Madison, NJ. It now seems that James did marry catholic. I will note that his sister, Fanny, had been married by an Episcopal rector. Unfortunately, the official entry has no additional data about the couple, so I will need to discover where the original catholic parish records are kept.