Hi, again.
Of course, fix the cooler (heat exchanger, in the radiator) before flushing so you don't contaminate the new fluid. But I can't comment on soldering effectiveness -- an aluminum structure is hard to solder properly, and I'd prefer to get a replacement radiator instead. But the decision is yours to make.

What I do when I'm flushing my transmission is do it two quarts (liters) at a time.
I've got the hose going from the radiator's upper connection (the line from the transmission to it is disconnected, while the lower line is left connected -- thus, fluid goes through the radiator's exchanger before coming out that upper connection). The transmission will pump fluid from it to the radiator along that lower line, and it would return from the radiator back to the transmission via the upper line.
That hose you installed on the radiator's upper nipple will drain the fluid into a clear plastic, 1 gallon, water bottle. I like to use a clear one, so I can see the color of the ATF coming out -- have the hose squirt the fluid on the inside wall of the bottle, so you have a clearer view of the color of the current outflow of fluid before it mixes with the fluid already collected in the bottle.

Start the engine at warmed-up (slow) idle, so the fluid will flow out the hose at about 1 gallon per 2 minutes (2 quarts or liters per minute or so).
When the gallon bottle is half full, I shut off the engine. Then I pour 2 quarts down the dipstick tube. Then I restat the engine and repeat this for the next two quarts, thus filling one 1-gallon bottle.

I continue further through more gallon bottles and many more quarts of fresh fluid, until the desired amount of fluid is exchanged (depending on the history of the maintenance and also the color of the fluid coming out.

I doubt you'll see any improvement in appearance until at least 12 quarts or so, and depending on the degree of contamination, you may need to run 20 or more quarts through before you're satisfied that the fluid coming out looks again like transmission fluid should look.

Nevertheless, the price of two cases (24 bottles) of transmission fluid is far, far less than a transmission rebuild! You may luck out and only require this flushing.

Best of luck.