Hi Rob,
Carvan and Bethel Chapels were daughter chapels of Henllan Amgoed Chapel, one of the earliest Independent Chapels in the area. As such their records would come under the name of the mother chapel, Henllan.
I think Bethel, Llanddewi was too late for you, being established in 1849. Carvan was established in 1804. Henllan records are on FreeReg, although I couldn't find a suitable David Davies.
He might have been baptised in Church, and converted later, so a look at the Lampeter and LLanddewi parish records might be worthwhile. I note that in the 1851 census he gives his DoB as 1805.
The Bishops Transcripts are on Family Search. Lampeter batch number C081741, Llanddewi batch no C081171. Remember they are indexed by the father's first name, so search using the first name David, and check the father's surname.
Regards,
Rhosgoch