Those temps you saw on that drive are totally normal when you're going up a grade in hot ambient temps at slower speeds. Overdrive off only helps with your trans temp if you're at a speed at around 40-45 mph or higher and the trans has shifted into 3rd gear. Speeds slower than 40-45 mph and the trans isn't in 3rd gear negates any benefit of turning overdrive off.
A trans cooler can only do so much. Even a pusher electric fan isn't going to make much of a difference. Those electric pusher fans actually block air flow over the cooling fins of the cooler when the fan isn't running.
The fact that you are monitoring your trans temp is good. That's more than half the battle. The key is knowing when it's prudent to pull over and give your trans a rest. The point I start considering pulling off and letting my trans cool is 230.
Steep slow speed ascents are always going to make your trans temp go sky high. The only way to avoid it is putting your transfer case in 4Lo. In 4Lo, you can climb steep grades all day and never overheat your trans. The only problem is, you can't really do this on pavement unless you have manual locking hubs like some of us do.
The amount of cooling provided by your cooler might be insufficient for driving steep mountain grades in hot ambient temps. My advice is pay close attention to your trans temp when you're climbing grades under the speed of 40-45 mph. When your trans temp gets above 230, stop, put the trans in park, and raise the rpms of the engine so the fan spins faster. Continue your drive after your trans temp has dropped below 200.
I read more closely through your thread and based off what you said, louder than normal transmission and now you're getting shuddering and throwing a P0770 code tells me your Hail Mary of dropping the pan and replacing the filter didn't work. I think you're definitely in the market for a rebuilt trans. Toyota doesn't sell them anymore unfortunately. Yota1 Performance might be able to sell you one.
Yota1 Performance Engine & Machine - Your Toyota needs all in one place! Since you live not too far away from them, you could drive there to pick it up and drop yours off to satisfy the core charge. Good luck!