The transmission fluid is cooled by the main coolant system, via a heat exchanger and transmission thermostat on the back of the radiator (below the expansion tank).
If this thermostat failed, the transmission could well overheat.
There's a few tools that will show you the transmission fluid temp - INPA is probably the easiest to use, followed by ISTA/D. Plenty of online tutorials on how to set these up. A Foxwell or similar scanner will also do it.
For INPA or ISTA/D on the e53, you only need a K-Line cable, like this: -
https://www.one-stop-electronics.com...usb-interface/
Re. the fluid change, at this mileage any fluid change is a risk but if it were mine, I'd drain it, open it up and replace the mechatronic seal, the bridge seal, the four jump seals, pan gasket and the filter. Then button it up and fill with new fluid.
I did this very same service, at about the same mileage as yours(~200,000kms). However, I chickened out and put the old fluid back in (along with a couple of litres of new ZF LifeGuard6, due to fluid loses from a leak and spillages when draining etc.). Made no real difference (for better or for worse) and about 80,000kms later, I did three fluid drain/fills, also with little effect (other than a few brief 5th gear glitches - which have since gone - phew!).
At nearly 300,000kms now and still working well (enough).
YMMV...