One of the first modifications I do to all vehicles I own is electric fans if they don't already have.
Can't speak for what size you need for a mower,but can say with certainty that a single 15" fan puts out enough CFM to cool a modified 350 chevy tahoe (even with nitrous and extremely high timing).
For an 10" or smaller fan look at trans cooler fans,they are slimline and usually around an inch thick or so.
If you would like to try somethign different,look for a couple computer fans. Usually they are about 2-4" and run on 12v,they are used pretty often on auto stereo setups to control heat.
I would ask if you've considered alternative methods though.
I use different types of internal engine coatings,and a big one right now is a heat dispersent that accelerate's heat transfer from the coated surface.
It's used on top of intake manifolds,intercoolers,supercharger cases and turbocharger compressors.
I've yet to use it on my mower or finish mower engines,but the concept is the same.
Coating the exterior and interior of the transaxle case,or just select area's that carry the most heat will defininately help.
Anyway,just a couple suggestions.
If you look at a couple truck stops you could find 12v fans of differing size's that could help also.
Good luck!