Someone asked a question about the banjo music back in 2010. I replied that Larry McNeely was named as the banjo player on the background music. Here's also credited as playing the steel guitar on the 1982 Dukes of Hazzard album, although I don't known which tracks he played on. Back in 2010, I could only find one Larry McNeely video on YouTube, but now there's more. It's not the exact music from the show, but some of the tracks have a similar feel. Here's the title track from his 1976 album 'Rhapsody for Banjo':
The track below is also on 'Rhapsody for Banjo', and should be spelled 'Honeysuckle Rose. Although it has vocals, I can imagine the Duke Boys driving to the instrumental parts.