There is only one Benjamin Griffins (or Griffiths) in the Bristol FHS index of baptisms 1754-1812. That covers the whole diocese of Bristol which includes Clifton and most neighbouring parishes.
SS Philip and Jacob
12.07.1778 BENJAMIN son of THOMAS & MARY GRIFFIN, BREAD STREET, LABOURER
The parish of Philip and Jacob is a fair way from Clifton, the other side of central Bristol.
Moving into Somerset, FreeREG gives one more possibility
Dundry
Baptism date 11 Sep 1788
Person forename Benjamin Symes
Person sex M
Father forename Benjamin
Father surname GRIFFIN
Mother forename Hannah
A bit further from Clifton, but nearer the expected date.
David