Ordained deacon in 1742 and priest in 1744, on both occasions by Thomas Secker, Bishop of Oxford, Barton held the following livings in the Church of England:[2]
He was elected Warden of Merton College in 1759, continuing until his death.
The Gentleman's Magazine considered Barton a humorous and cheerful man.[4] However, Bernard W. Henderson's history of Merton College comments that the college at the time cannot be said to have "displayed any great intellectual vigour".[5]