In the WWII era was there an organizational relationship between an "Engineer General Service Regiment" (EGSR) and a "Service Unit" ? I have items from both the 1303rd Engineering General Service Regiment and the 1303rd Service Unit. Are these items part of the same "1303rd" organization? The 1303rd EGSR was a temporary bridge building unit that served under General Patton's Third Army in the push from Normandy to Germany in 1944/45. I don't have any information about the 1303rd Service Unit except that it was based in Camp Lee, Virginia in 1943.
It is likely that both units you are referring to were part of the same EGSR Regiment.
The preface of the book Patton’s Fighting Bridge Builders: Company B, 1303rd Engineer General Service Regiment, by Joseph C. Fitzharris states that Company B was just part of the Regiment, and there were six other 'line' companies within the regiment. So there were other units out there that could have been at Camp Lee. Company B, however, was training at Camp Ellis in Illinois, as described in a web page concerning Earl Hall, the Officer in charge of the unit.
The cadre was formed in July of 1943, and grew to nearly full size at 1,700 men by March of 1944. The men trained by building bridges across the famous Spoon River in Illinois.
So whatever unit was present in Camp Lee, it wasn't Company B, but it might still have been another Company within the same Regiment. The Wikipedia article on Engineer Combat Battalions, which I believe is the current terminology, also mentions multiple companies,
Also known as "combat engineer battalions" (CEB), they were typically divided into four companies: A, B, C, and Headquarters and Service (H&S)
So besides the 'line' Companies, it may also have been typical to have a regiment Headquarters Company (Known as Headquarters and Headquarters in Army battalions.
I Have not been able to find which of these other companies may have been stationed at Camp Lee, but it could very well have been a member of the same regiment as Company B.