Dear Emmeline,
I really don't know, but I think that the certificate won't necessarily advance your cause much...as it won't link that Susan Carr to 'your' Susan Carr with any great degree of certainty...I suppose it really depends on how much you want to spend on certificates that may / may noy be a help. I suppose that if a Tricker is named as a witness, on the marriage certificate, it would certainly be a good sign.
Have you tried to track Richard Wiles and Susan Carr through the census records to try to eliminate them as possibilities, or found a subsequent death index for a Richard Wiles which might add weight to that possibility?
Is James shown on any subsequent census living with someone, such as to provide a clue, or might there any record of him having served time in the Army, Navy, or Police, or have emigrated (I guess not if you have a relative in the UK), for which something helpful might be extant?
I could not find the 1851 census record which you last mentioned.
Free BDM may not have fully indexed the years of interest yet, so perhaps something may yet turn up, or alternatively you might like to spend a penny looking at the 1837 Online site, if the BDM indexing has not been completed for the likely years. I repeat my earlier point...they may not have actually got married.
Sorry that I have not really been able to help.
Best Wishes,
Geoffrey.