The button is pre-CW (1840's) but possibly worn then too
1840's is absolutely correct! This style of button actually dates from the Second Seminole War and Mexican War periods. The 1830's-1840's eagle buttons are low convex (less domed), in comparison with the Civil War period examples. Additionally, the general service style eagle button (no letter for branch of service within the shield), was adopted for use as an enlisted men's button in 1854. Prior to 1854, the eagle buttons of both enlisted men and officers, carried the letter designation within the shield. Officer's eagle buttons of the 1840's were gilt and plated, (gold for artillery and dragoons, silver for infantry). Since there is no visible evidence of gilt on the example shown, the likely conclusion will point towards use as an enlisted men's artillery button (although there is the slight possibility that any traces of gilt have long been worn away, or obscured by ground action). During the Civil War, uniform buttons from earlier periods were in fact certainly utilized by forces both North and South, therefore Civil War use is always a possibility, depending upon context of the recoveries, and the history of the site.
The backmark appears to be a ring of stars, with possibly some lettering. This style of artillery button is consistent with the variety listed as AY206 in Tice's reference, although I'm not able to find an exact match for this particular backmark. From the photos, it appears as though one letter, a T or L is visible, and the other marks may be either stars or letters.
CC Hunter