I used to have a copy of the book of the film (or did the book come first?). The "duelling" tune is given there as "Wildwood Flower".

Wikipedia says the 1972 film is based on the 1970 novel by James Dickey who also wrote the screenplay along with director John Boorman, and played a Sheriff in the film.

Elektra Records had an album "New Dimensions in Banjo and Bluegrass" by Eric Weissberg and Marshall Brickman (1963) which was augmented with the "Duelling Banjos" theme tune played by Eric Weissberg and Steve Mandell and released by Warner Brothers. Arthur Smith's copyright claim was happily resolved, along with other residual rights claims on the earlier album resulting from unco-ordinated record company communications. Eric Weissberg explains the (involved!) history:-

here:- (It's a 2-page pdf)

Ross