Catalysis of the reaction: 2 ADP + cyclohexa-1,5-diene-1-carbonyl-CoA + oxidized ferredoxin + 2 phosphate = 2 ATP + 2 H2O + benzoyl-CoA + reduced ferredoxin.
cyclohexa-1,5-diene-1-carbonyl-CoA:ferredoxin oxidoreductase (aromatizing, ATP-forming) activity