I don't yet know whether he was at King's mountain. Here's what I have on him from a 19th-century book:
From "Descendants of Robert Campbell" by Rev. Charles W. Campbell, 1894. "Robert Campbell was born in County Down Ireland about 1718. Landed at Philadelphia with his brothers, when 7 years old, probably in 1725, but this is not entirely certain. Went to Prince Edward Co., VA. Afterwards thence to Rockbridge Co., VA. In 1776, he went with his sons to Carter's Valley in what was later the state of Tennessee, reaching the place where he afterwards lived, and where he died the day before Christmas. His death occured about 1810. His first wife was Leticia Crockett, who died in Prince Edward Co... "His second wife was Jane Allison, who survived her husband several years living with her daughter Leticia till 1820."
This is really odd for me because I went to the library to research Blair, found Turney, realized that Turney was in Hawkins county at the same time as Campbell. Blair/Turney are my dad's side and Campbell is my mother's. Her branch moved from Hawkins County to Claiborne to Franklin to Hardeman County. John Larkin Campbell, grandson of Robert, was there in Hardeman on the 1830 census and died there in 1870. John Larkin Campbell's grandson, William Allen Campbell, moved to Arkansas in 1900 or so, and was a doctor. He settled in Phillips County in the Delta.
I have more info (and a few pictures) on my website:http://surgery.uchicago.edu/hblair/