If the shaft on the briggs is the same diameter as the kohler take the pulley setup off the kohler and put it on the briggs.
These can get stuck on the shaft after time. Look at pulley for a set screw that locks it on shaft. (if it has one) .It may take a pulley puller to get it off there. spray real good with WD40 or something. Sometimes it is easier to order another pulley.