My transmission cooler lines run through my radiator first, then through an aftermarket transmission cooler and back to the transmission. Would it be best to bypass the radiator and just run through the aftermarket cooler?  

Absolutely not the radiator is where you get the most cooling for the transmission fluid unless your cooler comes with a fan. If not most of your cooling will come from the radiator a good cooling system like a aluminum radiator, electric fan, and good hoses is all you need to keep the transmission fluid cool, the transmission cooler is just a little add on.