Dirk,
x2 on the "all auto transmissions have a cooler" at least in the radiator tank.
One pretty sure way to tell whether you need a cooler would be to check the trans temp.
The temp sender needs to be in the pan. In my C-6 it is located horizontally at the rear of the cast aluminum pan just below the modulator. Don't put it in the bottom of the pan unless you want it to get sheared off
On the street the temp in my automatic rarely goes above 180 even after a combo of freeway and traffic driving. On the track it sometimes gets to 280 depending on the track and how aggressively the car is being driven. No problems yet...
The operating temperature of an automatic is dependent on a number of factors besides usage. In no particular order they include the size, type and stall speed of the converter, any restrictions in
oil flow inside the trans, inside the converter and/or externally, the type of fluid in the trans, the shift timing in the valve body and the type of friction material in the bands and clutches. I may have omitted something but you get the idea
Check your temp. Then loop the external cooler hose so it bypasses the cooler, and you may get an idea about whether you need a cooler at all.
Oh, and do this on a really warm day....
Tom