Sounds like once the trans oil gets hot you start losing drive. The hotter the oil, the thinner it gets, and the more can get by worn pumping parts.
Changing the existing oil and replacing it with regular JD Hy-Gard hydraulic fluid (not the low-viscosity version) may get it back in the game.
Dealer probably can't reproduce it as they are not actually driving the machine, they are probably just letting it run for an hour which won't build any heat in the trans.
Go for the oil change.
Yup, that's what I'm leaning towards as well. At least one dealer said they did drive and mow it when they took it for troubleshooting, but I have my doubts about that now. When I mowed just yesterday in 90 degree heat, it started after about 30 minutes(sooner than usual, and got much worse.
Hy-Gard it is! Thanks!