You're confused, GUEST Q. What Malcolm said was:

It wouldn't be at all surprising if On Christmas Day derived from the homilectic chapbooks that circulated widely in the earlier part of the 19th century; as, it appears, did The Holy Well, whatever its antecedents may have been.

Malcolm thus did not attribute The Bitter Withy to a nineteenth century chapbook, but The Holy Well, a similar but different song, and On Christmas Day, a totally different song.

I don't know what you mean by "in its form" it could be from that era, though...

Its appearance in an 1836 book of carols is likewise irrelevant; The Cherry Tree Carol, St. Stephen and Herod, etc, are considered both ballads and carols, and Child included them in his collection. So just because it was in the book of carols there would be no reason for Child to exclude it.