The "cooler" in the factory radiator performs one function. It maintains the trans fluid's temperature. When it's hot it cools it when it's cold it warms it up. Transmission do not like to run too hot OR too cold and too many don't realize that the "cooler" is also a "heater" when needed.

If you choose to run an additional, external cooler make sure it has a thermal bypass that will not allow it to cool the trans fluid when it's already too cold.

So, no, do not run just an external cooler.