Yes, a bishop (Anglican) can impale his arms with those of his see but in doing so replaces the helm and crest with a bishop's mitre and usually accompanied by a crozier and/or processional cross.

If these arms had, as you previously mentioned, a helm and a crest, they most certainly weren't ecclesiastical arms of office and even more against such a display would be the fact that I believe that Bishop Stacpoole was of the RC faith which forbids any of their clergy (of any rank) to bear a helm and crest.

From what I have seen so far, I would be pretty certain that these are the impaled arms of an armigerous married couple with the wife (having a maiden surname of Stacpoole) not being an heraldic heiress.

To get a familial connection, you are going to have to find a Stacpoole wife somewhere because it is her arms that are depicted on the sinister.

If they ARE ecclesiastical arms of a RC bishop, they wouldn't have been recognised or recorded in Britain as the State Religion is still Church of England and the College of Arms cannot recognise RC ecclesiastical sees, even though they do grant personal arms to RC bishops.

The fact that they possibly come from the maternal side of the family (as yet unproven) precludes any present day use of those arms as arms are only inherited through either the male line or through an heraldic heiress when the arms would be quartered with the armigerous husband. If the husband was not armigerous, the arms died out on the death of the last surviving heraldic heiress, presuming that there were other sisters who would each have been equal heraldic heiresses).