"Brigge" is going to have different spellings (see Briggs in the Yorkshire Visistations) although in my quick scan none of the variations seems to be in the Yorkshire Visitations as having arms.
In the context of that section in the Wakefield Manor Court Rolls, "Grave" is probably more commonly termed "reeve" (is that spelling a Yorkshire variation?) since they were working for the bailiff and Reeves were elected peasants who helped the bailiff and steward manage the estates.
I am pretty sure a dispensation was priced accordingly to your social status but was not cheap. Plus, it could be any sort of 4th degree of relation through her (or his!) first spouse so they may not have had a common gggf/m. There's a lot of social activity which seems weird to our modern sensabilities - the wealthier lower classes may have sought out dispensations because that was what the gentry did even if they weren't considered of that class.
Lower class people didn't have a ton of records & this family may have enough wealth to know about their relations and buy dispensations but not enough status to own manors or afford (or need) lawyers. Ultimately, it will be very difficult to prove the line since there's no pedigree available.
