The rose and briar verses are not part of Barbara Allen (Child #84) in Child, but they certainly are present by the time Bronson was collecting tunes - in 67 of the 198 versions of Barbara Allen he found.
The rose and briar verses were in
Earl Brand (Child #7),
George Collins (Child #85),
Fair Janet (Child #64),
Lord Thomas and Fair Eleanor (Child #73),
Lord Lovel (Child #75), and
Fair Margaret and Sweet William (Child #74), and mentioned in the original post.

Child #65 Lady Maisry/Susie Clelland has the verses in Bronson, but not in Child, as does one version of Matty Groves, Child #81.