WELCOME to LMF...
As you have discovered that trans-axle will not tolerate a heat load for long...
The first symptoms begin when the hydraulic oil cooks and looses it's ability to lubricate the internal parts... then as it is forced to operate it will cook internal seals and finally destroy itself....
I don't think you can add external coolers OR a large oil reservoir to that lawn tractor... but that is what is needed to operate that machine in those harsh conditions...
The trans-axles on Many lawn mowers are not designed for easy service... there is nothing that is completely Non-serviceable, some things are just designed to be easily serviced... others very difficult...
If you sill have any warranty... have another replacement and then either sell it or use it on a flat area...
If your warranty is up... you have only a couple choices...
Try to do an oil change first... Use a good synthetic engine oil 5W 40, or 15W 50 ... in the trans-axle... Your first challenge will be getting most of the cooked oil out....
fill it with fresh "Synthetic Motor oil" Mobil1 15w 50 OR Rotella T6 will work fine... if the internal parts are not too damaged, you will have an operational mower again... BUT don't use it on any continuous hill climbing job... .... Good Luck, you need something better designed to take that kind of load & get rid of the heat...
:smile:KennyV