We've now gone through the original Carter Family songs from A-F. See the details in the following posts. I'll put an asterisk by the songs that they wrote or have a unique arrangement. We can see the Carters songs are based on other existing songs or fragments of existing songs.
Amber Tresses: 1874 "Amber Tresses Tied in Blue," Words Samuel M. Mitchell, music H. P. Danks.
Anchored in Love; 1911 "Anchored in Love Divine" James Rowe & James Vaughan
*Answer to Weeping Willow: a rewrite of "Bury Me Beneath The Willow"
Are You Lonesome Tonight?: 1926 Lyrics Roy Turk, music Lou Hindman.
Are You Tired of Me, My Darling?: 1877 by Cook and Roland
*Away Out on Saint Sabbath: a rewrite of Bury Me on the Lone Prairie
*Bear Creek Blues; from Leslie Riddle, traditional blues verses from Blind Lemon others
Beautiful Home: 1898 by J. Howard Entwisle and Johnson Oatman
Beautiful Isle O'er the Sea: based on an earlier song, maybe Stoneman's.
Behind Those Stone Walls: based on an earlier song.
*Birds Were Singing of You: A.P. Carter as far as we know.
Black Jack David: traditional; taken from Cliff Carlisle; David Myrick
*Blackie's Gunman: rewrite of an unknown song
Bonnie Blue Eyes: arrangement of a traditional song
Bring Back My Blue Eyed Boy: arrangement of a traditional song
Bring Back My Boy: same song as above- arrangement of a traditional song
Broken Down Tramp: rewrite of an earlier song
Broken Hearted Lover; arrangement of a traditional song
*Buddies in the Saddle: 1940 attributed to Maybelle Carter
Bury Me Under the Weeping Willow: 1909 traditional folk song
*By the Touch of Her Hand: attributed to A.P. Carter
Can the Circle Be Unbroken (Bye and Bye): 1907 Ada Habershon, Music: Charles Gabriel.
Can't Feel at Home: 1919 as far as we know, traditional folk hymn and spiritual
Cannonball (Blues): from Leslie Riddle based on earlier song
Carter Family and Jimmie Rodgers in Texas; skit
*Carter's Blues; 1929 based on another song; title would need to be changed to avoid copyright issues.
Charlie and Nellie: traditional- based on earlier recordings
Chewing Gum: 1925 traditional song
Church in the Wildwood; 1857 James Rowe- words, William P. Pitts- music
Coal Miner's Blues: arranged collected by Leslie Riddle; traditional lyrics from coal camp.
Cowboy Jack: 1928 traditional based on an earlier song
Cowboy's Wild Song to His Herd: based on an earlier song
*Cuban Soldier: based on an unknown song
*Cyclone of Rye Cove: based on a poem sent to AP.
Dark and Stormy Weather: based on traditional lyrics
Dark Haired True Lover; 1918 text is in Robert Gordon Collection #1536
Darling Daisies: 1882 "Down by the Garden Wall" by Max Vernor.
Darling Little Joe: 1866 by V. E. Marsten.
*Darling Nellie Across the Sea: based on an unknown song
Diamonds in the Rough; 1897 CW Byron words and LL Pickett
Distant Land to Roam; LM Bandy's 1902 song "Leaving Home"
Don't Forget Me Little Darling; 1874. C.W. Vance- Words and RS Crandall
Don't Forget This Song: 1910 based on "Bad Companions" or "Young Companions"
Dying Mother: 1881 Nona Lawson- words and C. M. Tate
*Dying Soldier: based on an unknown song; the title may be used
East Virginia Blues: Traditional
East Virginia Blues No. 2; Traditional
Engine 143; 1915 traditional ballad
*Evening Bells Are Ringing: 1934 based on an unknown song by A.P Carter
Faded Coat of Blue; 1865 Civil War song by J.H. McNaughton
Faded Flowers: 1851 song by James Powers and JH Brown
*Fate of Dewey Lee: based on a poem by A.P. Carter
*Farewell Nellie: reworking of traditional material by Sara Carter
Fifty Miles of Elbow Room: written by Herbert Buffum 1879-1939
Foggy Mountain Top: traditional, based on earlier song
Fond Affection: traditional arranged by Carters
Forsaken Love: traditional arranged by Carters
Funny When You Feel That Way: 1873 "It's Funny when you Feel that Way" by George Harris
By looking at the list of 59 songs above we can see there are 15 songs that would be difficult to use freely without getting permission. Several of questionable songs you could use but would need to change the title: Bear Creek Blues (could be Cripple Creek Blues or whatever) since the Carters (Leslie Riddle) used traditional lyrics.
It looks like there are 6 songs that they wrote (four by AP). Even the ones they wrote were based on other material. Some of the other 15 questionable songs are based on songs but we don't know yet what the songs are. The Carter also received songs from their followers. I don't know the extent of songs they got except the "Fate of Dewey Lee."
The amount of time and effort AP took to find their songs should not be underestimated. You need to remember that many of their arrangements are copyrighted (except the ones they based on other earlier arrangements like "Black Jack David" and "Charlie and Nellie.")