Homeless, if you & these gents are playing for folks who enjoy the songs as they were performed by the artists you mentioned, I would suggest that they would most likely prefer the sound of the bass lines as the recordings were done. The bass line, which is sparse in most of the songs such as "Cold, Cold Heart" can help to define the sound of those songs.
However, bearing in mind that Western Swing (just outside of Tulsa, where my Daddy grew up with the Wills brothers it was pronounced "Western Swayng") was the "country folks" answer to the Swing/Big Band of the time, have some fun with the bass lines in those songs.
In those decades of the 30's, 40's, & especially the 50's, it is sometimes difficult to define exact genres for songs since the American popular Music genres evolved from the same roots--the merging of Music from various immigrant groups. For example, Bob Wills is considered "Western Swing," but in a few of the songs Bob Wills recorded, one can hear the beginnings of what was to be called "Rock n Roll." Such as, when the whole band was singing from the bandstand, "Everybody let's rock, rock, Everybody lets roll, roll . . ."