I know you have removed the top shroud, but what about the cover tins over the cylinders to make sure they are not packed full of dirt and chaff? I've seen too many have a clean topside and the fins were packed solid.
As I recall, B&S had a number of overheat issues with this design.
Next idea is to go up one size on the carburetor main jets so it will richen the mixture slightly and help cool the combustion slightly. These engines were tuned to run super lean due to emissions imposed on the industry.
As for the transmission(s) they have cooling fins and usually cooling fans. The fans are common for rounding out the spline teeth in the center and gradually slip and lose cooling efficiency.