I know the feeling, I recently had to replace my 41Te.. but I've gotten a little smarter since...
Here's an interesting note about how your A604 operates
The way I heard it, when Chrysler made the 3.3/3.8, they did one of two things when they made it so only the A604 went with the 3.3/3.8. One possibility was the whole thing was a mistake, and the A604 should not have left the production lines until it was perfected. The other possibility was that they looked at the 3.3, and found its only weakness was with the crank and rods. So the stuck the weak A604 on every 3.3/3.8- thus ensuring the engine will never receive damage from mods, etc. Basically the A604 acts as a pressure valve and takes all the heat and damage- and not the engine's internals. The A604 will fail many times before the 3.3 will ever even get a hint of damage.
The result it the ability to add almost any reasonable amount of boost to the 3.3, and it will be fine (so long you don't go lean and use cool enough air). Once power gets too high however, the A604 will go- thus leaving the engine untouched (not bad.. cheaper to get an other A604 from a yard then to rebuild a 3.3/3.8.. and easier to swap trannies then engines!).
Basically that should answer the question, yes heat will kill the tranny but it's not climate heat that is the problem, it's engine heat.
Heat causes the fluid to break down, so most importantly you need to flush the tranny every 15,000 miles (Full flush, filter replacement, and refil with Mopar ATF+3 ONLY!, don't let a shop use Dextron III it'll kill your tranny)
Also you may want to have a shop install a drain plug in your trans pan so you can drain around 6 qts and replace it every other oil change or so. The tranny should hold either 10 or 12 but the TC itself holds about 4 qts of fluid so you won't be able to drain it all.
Next look into getting a tranny cooler, and maybe even a shift kit Trans-Go makes a shift kit for your tranny, in fact I'm running the 42Le shift kit on my 41Te A604.
With these transmissions you want quick hard shifts (as funny as it may feel) not smooth fluid shifts.
here's some more info you may want to read.