As is so often the case, the Ebay seller is WRONG. This type, which has a "vertical rim" and tabs on the edge of the rim for attaching to the leather horsegear is Spanish-American War era (1890s) to World War One era and later.
The civil war version is MUCH thinner, lacks a "vertical rim," and has a lead-solder filled back.
By the way, note that the Civil War In Tennessee website does not actually say the bridle in their photos is a civil war one. It just says "US Army bridle."
And as is also so often the case, Creskol is correct.

It is an "accouterment," not a rosette (which has a bar on the back for leather straps to pass through).
The photos below show the SpanAm-War-and-later version, on original leather.