Are you towing in 4th gear or 3rd? I ask because your post makes it sound like you are towing in 4th. That's bad at any speed with the older 4 speeds. With the 4 speed, TH mode only changes the shift pattern from a dead stop. It doesn't control the shifting between 4th and 3rd or prevent OD from being used. OD kicks in at 40 so if you are in 4th and going between the speeds you were it doesn't surprise me that it heated up. At those low speeds it doesn't take much of a speed change to force it to downshift into 3rd. Constant shifts between 3rd and 4th and back again will generate a lot of heat.
Are you using Dex VI when you do a fluid change? It's recommended for your truck even though it had Dex III in from the factory. Dex VI came out in 2006. Its a synthetic blend and runs far cooler than Dex III did.
A trans can take short bursts above 220 without permanent damage. But 300? I dunno. That's a scary high temp.
EDIT: If you add an auxillary cooler, don't bypass the factory cooler, put it inline.