Wither’s Artillery was the 1st Mississippi Light Artillery. There were ten or eleven batteries (companies) and they never fought together as a regiment. Among other places, they fought at Vicksburg and Port Hudson (where my great grandfather served).
See http://www.mississippiscv.org/MS_Units/1st_mississippi_light_artillery.htm
There is no record in the Civil War Soldiers and Sailors System on the National Park Service website (http://www.itd.nps.gov/cwss/soldiers.cfm) of a W.R. Lee. There is a William J. Lee who was a member of Richards' Company, Mississippi Light Artillery (Madison Light Artillery). This unit was from Madison County.