Hard to know where to start.
First off, 203 degrees on modern trans fluid is nothing to be concerned with. I wouldn't waste the mental energy to give a ****, never mind
spending time, money, effort, and enthusiasm to "fix" what isn't a problem. GM doesn't turn on the electric radiator fans until 220+ degrees, so the trans fluid cooled by the radiator is going to be ~200 degrees on my wrong-wheel-drive cars.
Second, there's three ways to "add" an auxiliary trans cooler. Before the in-radiator cooler;
After the in-radiator cooler; and
Instead of the in-radiator cooler.
Which way is best depends on the circumstances. Around here, I'd want the radiator to heat the fluid before it's returned to the transmission. (I've never bothered to see if that's how the aux. coolers of my trucks are plumbed.)
Found this >>
https://www.klclutch.com/torque-converter-fluid/how-do-you-drain-a-torque-converter/
Step 4 for method A. Figuring if I pump 5.5 litres through then that's all the old fluid out.
5.5 liters of fluid is
nowhere near enough fluid to flush the trans with. IF (big IF) you
already drained the pan and installed a fresh filter,
and then refilled the pan, 5.5 additional liters MIGHT be enough to flush the converter with. Probably not.
I just finished changing fluid on my '03 Trailblazer 4L60E two days ago. I dropped the trans-to-cooler tube at the radiator, (because it was easier than the cooler-to-trans tube) pushed it into a two-foot rubber hose leading to a BIG drain pan, and started the engine. As soon as the fluid coming out of the hose stopped gushing and began trickling, I knew the pan was nearly empty--the filter was sucking air. This does not empty the torque converter, it empties the pan.
Remove pan, change filter, clean pan and magnet and pan gasket surface area of the case, install pan with new gasket.
I put 8 quarts of Dex/Merc ("Dexron III compatible") down the dipstick tube (Usually I only install 5 quarts, but this is a deep pan), opened five more quarts, and started the engine. I dumped fluid down the dipstick tube as fast as I could, until the fluid coming out the rubber hose into the drain pan was bright red. Shut off engine, reconnect cooler tube, start engine and top off fluid as needed.
I went through about 14 quarts (more than 13 liters) altogether, but now all the fluid in the trans is fresh except the little bit trapped in the clutch piston housings, servo, accumulators, etc.
I could have disconnected both cooler tubes, regulated my compressed air supply down to ~30 psi, and back-flushed the cooler with compressed air--but I didn't. There was no debris in the pan other than some fuzz on the magnet, so I figured there'd be no debris in the cooler, either.