As far as the trans goes 200f aint that bad... But I would invest in adding a second cooler in line with the factory cooler..
I installed the biggest cooler I could fit...And lowered the factory cooler to get it out behind the chevy bowtie on the grill.
Here is pulling the smokey mountains. 230f coolant temps, but 175f tranny temps..
1st gear, 3500 rpms, 30 mph....lol... Hey its only a 4 speed!!!!!
Just a note: I could easily speed up.. But chose to pull these hills sitting @ 3500 rpm.. Its a good power band and the 5.3l will pull all day I believe.
Ran with A/C on the whole trip... The A/c on keeps the fans on. helps cool better.
Video...
https://www.flickr.com/photos/93289479@N04/35822079640/in/datetaken/