I find one Confederate and three Yankees by that name in the American Civil War Research Database:

Benjamin F. Carter

Enlisted as a Corporal
"C" Co. MO 6th Infantry, CSA

I do not find his record in my CSR files

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Benjamin F. Carter

Enlisted as a Private (date unknown).
Intra Regimental Company Transfers:
* from company C to company F
He also had service in:
"C" Co. MO 1st SM Cavalry USA

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Benjamin F. Carter

Enlisted as a Private (date unknown).
* from company C to company F
"C" Co. MO 1st SM Cavalry USA

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Benjamin F. Carter

Enlisted as a Private
"K" Co. MO Home Guard Infantry USA