Not sure if one would work for you all but an automotive transmission cooler can be had at a junk yard for next to nothing. Mount it to the mower somewhere that gets moving air and plumb it into the oil system. We used to use power steering coolers on Harley's back in the day as a cheap and easy oil cooler. Maybe a 12 ducted fan operating off a temperature sensor to move more air across your oil cooler?
Tom
Tom