The only reference for Bunting I could find for you is taken from Old Belfast Families and The New Burying Ground, Graveyard Inscriptions Vol. 4, compiled by A.C.W. Merrick and edited by R.S.J. Clarke.

BUNTING: Erected in memory of John Bunting who died 18th Oct. 1828 aged 52 yrs. Harriet Bunting died 12th Jan. 1830 aged 23 yrs. Thomas Ash Bunting died 17th Nov. 1833 age 21 yrs. Mary Bunting died 17th Jan. 1837 aged 21 yrs. Sarah Bunting died 21st May, 1837 aged 67 yrs. The rist and last on this tablet were the parents of the intervening three, their beloved and only children.
[John Bunting ran a school for young ladies at 1 Donegall Place. In the 1800's it was situated at 67 Donegall Street. His brother, Edward Bunting is famous for having chronicled so much of Ireland's old music]

Hope this helps,
best wishes
Maree