Since the Catholic priest’s name is not on the marriage license, it’s safe to say that he did not officiate at the wedding.
At the time, you needed a dispensation from canonical form. If this happened, the marriage should be recorded in 4 places (some of these might overlap, for example if #1 is also #4):
- Your parish of baptism.
- The Catholic parish in which the marriage occurred. Remember that a Catholic parish is a geographic location, not a building. Therefore, even if it was in a non-Catholic building, it was still within a Catholic parish. That parish is obligated to keep a record of the marriage, even though the officiant was non-Catholic.
- At the Chancery of whatever bishop granted the dispensation. This might be the diocese where the ceremony happened (more likely) or the diocese where you, as the Catholic party, resided at the time (a little less likely). Either is a possibility. This would be the most difficult to locate, but at the same time, the least likely by far to misplace the file or fail to record it.
- The parish where you did your pre-marriage preparation. They should have a pre-marital investigation file. It would not (not likely) be in their sacramental record books, but in the marriage files.
You might find that #1, 2 or 4 has nothing more than your wife’s name and a protocol number, like 1984-0238 which would refer to the document granting the dispensation. If you find that, it would be easy for the Chancery to locate the dispensation in their own files. If the Chancery has those files recorded on computer they might be able to do an easy search.
Having said all that, the starting place would be the parish where you were baptised. Of all the possibilities, this would be the best place to start.