In NO WORD OF THEM I have culled as much historical data as I could find which addresses why this unit of New York State Sharpshooters has been confused since at least the late 1890s as being part of Berdan's United States Sharpshooters. Last night I came across this comment: "Berdan assumed command of J. H. Hobart Ward's brigade when the latter was wounded, and led it throughout the rest of the campaign, as well as during the subsequent Bristoe and Mine Run Campaigns."
This is perhaps another source of confusion, since the First Battalion New York Sharpshooters were engaged at both Bristoe and Mine Run. They were organizationally part of Gen. Lysander Cutler's "Iron Brigade", not J. H. Hobart Ward's brigade. The OR does not indicate that the NYSS were ever under the command of Berdan.
By the way, in the 1850 US census, Hiram Berdan, of New York, was located in Kenosha Wisconsin working as a machinist.