340 coolant temperature

The tranmission cooler is built into the Champion radiator bottom (that is where the pipes go to).

if you have a high stall speed torque converter, you need an auxiliary trans cooler too. The one built into the radiator is merely good enough for a stock stall speed unit.

On the other hand, you never said if this car has a high stall speed unit or standard. Since you apparently have no overheating problem at high RPMs, then it is probably just a fan/shroud issue. Good luck!