A Phineas Bateman who could be James' brother is the innkeeper living in Fleet, Lincolnshire in 1841. He is aged 80, and not born in Lincolnshire.
This is likely to be the same Phineas Bateman who married Elizabeth Rignol at Long Sutton, Lincolnshire in 1798.
This marriage appears on the familysearch site, and also on FreeReg - but unfortunately these don't show details such as witnesses, or whether the groom was from another parish - so it might be helpful to see the original register entry.
Phineas died in 1842 aged 83 and was buried at Long Sutton. If age at death is accurate, he would be older than James, and it is therefore possible that he was named after his grandfather (although of course you can't rely wholly on traditional naming patterns).
It may be relevant that James named his first son Henry. There are not many Bateman burials in Ramsey during the second half of the 18th C, but Elizabeth Bateman wife of Henry was buried there in 1770, and Henry Bateman in 1781.
Some Ramsey families seem to use the parish church at Doddington, Cambridgeshire sometimes - I'm not sure if this was because they moved back and forth across the county border, or if it was just a preference. However, I don't think that the records are available anywhere online for the mid 1700s.