The 1836 publication of this atlas was the 1st edition. Virtually identical maps, with the exception of the publication information, appear in subsequent editions and atlases by Mitchell and others. The publication statement and the color palette definitively tie this to the 1836 publication.
While this is most likely from the 1838 edition of Bradford's atlas and Walter's notes indicate that, the color palette doesn't match other copies available online that are defintely from that atlas. However, the colors also vary between those maps.
Different sources have different publication dates for this atlas. The Library of Congress uses information from the supplementary index to date it to 1843. This is a fine steel engraving typical of Archer's work.