Ok Matt. Time for some unconventional wisdom here. You've rebuilt the carb, right? Do you have fire at the plug when it shuts down? Do you still have fire? If so, here's where I'm going to direct you.
I've just finished going down the road you are at right now with a 1200 Cub Cadet. We have pretty much the same machine. Mine is a straight drive, your's has the hydro. Ok?
Something I found out: This line of Cub Cadets has that dang heavy muffler hanging off the front of the engine. It has a tendency to break at the mounting points. One of those is in the air duct to the right of the tractor. The other? It attaches to the exhaust pipe on the left side of the tractor.
I cannot tell you how many used mufflers I looked at on ebay that were cracked right around the mounting for the input exhaust port. I finally ended up buying a brand new muffler from C&G. Best money I spent on this tractor besides getting the engine rebuilt.
Now then, with the exhaust flange being broken so easily, this has a tendency to BURN
the exhaust valves on these engines.
The exact same things you are speaking of with running for a short bit then shutting down were happening to me. I rebuilt the carb, flushed the tank, had great spark, the whole nine yards. It ran fine cold, but when it just started to heat up, you couldn't get it to run worth a flip.
I yanked the engine and sent it to a machine shop for a rebuild. The machinist called me back and had me drop by the shop.
The exhaust valve had a burned flat spot that faced the exhaust pipe! As it heated up, the valve would expand and stop sealing properly. No seal, little to no compression, no pulling fuel properly, and it acted like it ran out of fuel.
Pull your hood, your front grill, your cooling tins and then the muffler. Check the input on the muffler that the exhaust pipe goes into. Bet you a RC cola and a moon pie it is cracked just like all the rest I've seen. On top of that, I bet you have a burned exhaust valve.
Remember, these tractors have been driven into the ground with little to no proper maintenance. They also are prone to excessive vibration and this lends to sagging ISO engine mounts, bad drive trains (yes I dealt with that too), cracked mufflers, and who knows what else.
On a positive note, I drove mine around the first time yesterday with the aerator as a test drive after my restoration. My 18 HP Simplicity doesn't have the power my 12 HP Cub Cadet has and the Simplicity is about 20 years newer!
Don't pull your hair out. My head is bald enough for both of us. It'll take you all of about 30 minutes to pull it down as far as I've suggested if your bolts come out as easy as mine did. I'm glad I went into it as far as I did. Now, I'm into the testing and re-tightening phase.
Oh! Don't run your Cub Cadet without a muffler. You are only asking for trouble - as in burned exhaust valve.
I love my Cub Cadet!
Let us know what you find out. When I get time, I plan on doing a write up with photos of what I ran into in the rebuild of my Cub Cadet. It's the least I can do after the good folks here helped me out like they did.