A birth certificate for one of their children would confirm the mother's maiden name, then you could get the civil record. Cost = two certificates, but will provide the most solid information.
The most likely entry in freebmd, presuming the marriage took place in the Dewsbury area, is the 1870 marriage to (potentially) Mary Dwyer. You could first see if you can find an appropriate Mary Dwyer of the right age born Dewsbury, then order this only if there it looks possible she was your Mary b. 1851.
Catholic parish records are often difficult to locate, and in many cases have not been handed over to archives but are still with the diocese. Also, in the case that only one spouse was Catholic, you are likely to find the marriage was at the registrar's office, in which case there's no parish record to be found at all.