Bearss's The Vicksburg Campaign in three volumes refer to a Cavalry Battalion and also a Sharpshooter's Battalion

Bridges' Arkansas Cavalry Battalion, Maj. H. W. Bridges June 22-July 4, 1863

Bridges Sharpshooter Battalion

Also, on page 722, Vol. III, in a footnote, "A section of the 3d Maryland Battery had joined Maj. H. W Bridges' battalion at Fish Lake, near Greenville, on May 1."

There is no mention therein of his death.

The U. S. CW Research Database list him as following

Henry W. Bridges

Enlisted as a Major
Field & Staff AR Briders' [sic] Battn SharpShooters

See also:

http://freepages.family.rootsweb.ancestry.com/~bowen/legion/aboutactions.html