Excellent write up and very informative, the exact same thing has just happened to my LC200 under very similar circumstances, ie towing, high ambients and after running for very long distances / hrs.
Of interest - I limped the last 150 k into Tennent Creek across the Barkley, the tank was just above 1/4, I fuelled up with the intention to stay the night and buy a secondary fuel filter in the morning, once the tank was full the car went perfect again, my thoughts where the 'cool' fuel had the tolerances in the fuel pump improve.
I also received unsatisfactory results from the Toyota service dept in
Darwin where basically no form of corrective measures had been implemented, they only said the cats are now faulty.
I'm thinking of trying to get non - oem high flow cats installed, and as you have experienced, have the fuel pump replaced and the tank checked for cleanliness.
What I find disheartening is the fact that I wont have the faith of it being reliable on the next big adventure.
I wonder if many other petrol LC200 owners have experienced this?
Feel free to mention my case to Toyota if required.
Kind regards, Scott.