There was no official statewide registration of births and deaths in Tennessee before 1908. Four cities did keep some earlier birth and death records (Chattanooga, Knoxville, Nashville, and Memphis), which are available through the Tennessee State Library & Archives.
This means that finding a more specific date of birth for Bedford Thomason may be a bit more difficult. Perhaps you might find it listed on his Grave Marker. Have you tried findagrave.com ?
I have been unable to locate a death record for Bedford, which would likely give you a date of birth - although this is considered a secondary source and might not be accurate. I do know that he lived into the 20th century because he and Jennie are enumerated in the 1900 Census in Ozark County, Missouri.