Hi,
I know they are brothers. I have done some research on the family and the census will show that.
John L. F. Bronson and Louise Salyers 4 children, Charles, Carrie as you mentioned, William Morgan and your Wade Hampton Bronson. All children born in Ky., William was a year older than Wade.
John died of course before 1900 and Louisa in 1937.

Beth