Think Barr Academy was a different location. Just found an article dated 1797 of a robbery at John Mayne's schoolroom at Snails Green, Great Barr.
John Mayne was still at Barr School/Academy in January 1800. He took over the Barr school in 1788, having previously had a school at Rotton Park from 1786 . So this could quite possibly be John Lichfield Mayne's father. One advert in 1798 announces "John Mayne and Sons Barr School will, open ....... ". So John Lichfield Mayne, assuming he was one of the sons, may have been at Barr School until at least 1798. By 1800 adverts quote "John Mayne and Son" at Barr School - singular. This would have been John Lichfield W's brother, William (who later took over Barr School from their father).