Certainly one idea that pops into my head is to write a short filter-script which concatenates the continuation-lines into one contiguous string. It reads line-by-line, noting whether the current line is a continuation, if so appends it to the stashed line, otherwise outputs the stashed line (if any) and stashes the current record.
If you apply this filter to both files in turn, you have now reduced the complexity of the problem considerably because continuation-lines are no longer a concern in the filter-output files. Now, maybe you can apply tools such as diff to them, and so on. I’m really warming-up to that idea, as I describe it.