Yes, running two coolers will definitely help lower the temps, but at the same time, keep the trans fluid running too cool in colder weather. If it was just a case of running too hot, I'd probably have chosen to run a big cooler and bypassed the radiator cooler too. But like I've already stated, trans fluid has an optimum operating temperature range and running too cool will allow moisture to build inside the transmission which isn't good.
Think of it this way. Why do we have a thermostat for the engine? It's because without it, the engine would take longer to heat up to an optimum operating temperature. The thermostat keeps the coolant running through the engine until it reaches 180 degrees and then opens to cycle the coolant through the radiator to be cooled and returned to the engine. By the same token, the stock trans cooler in the radiator shortens the time it takes the trans fluid to reach optimum operating temperature because the cooler is bathed in the hot coolant and transfers heat to the trans fluid via conduction.