Does your transmission fluid smell burnt?
The engine coolant system moves a lot of water and displaces a lot of heat.
The amount of thermal energy you would need to put into the system by using the radiator mounted transmission cooler to drive the engine coolant up should burn the transmission fluid in short order.
Perhaps check other sources of water temp increase before condemning the transmission.Perhaps the fan clutch first?
It will help definitely to establish that the engine's cooling system is functioning properly. Considering the lesser cost of two testing options, without actually beginning any major transmission work, I would temporarily mount a fairly good-sized oil cooler in front of the radiator, then
disconnect the cooler lines from the car's radiator and re-route them to the outside, air-cooled coil. This removes the question of heat being dumped into the engine's cooling system. Drive the vehicle thusly awhile, and see if the engine temp. remains where it should be. If it
does, suspect the transmission as contributory. If it does NOT, then something is afoul in the engine's cooling ability itself.
True, eng. coolant can remove a lot of heat from trans. fluid, but keep in mind that the trans. fluid temp. MUST be higher than the eng. coolant temp., to effect cooling of the transmission. Otherwise, the trans. fluid is COOLING the eng. coolant. The important thing is the TEMP. DIFFERENCE between the two media, let's say, 195` eng. coolant "cooling" 215` trans. fluid, a "Delta-T" of 20`. With the air-cooled coil outside the radiator, ambient air, let's say at 80` is cooling the 215` fluid. Which situation do you feel is more effective? Yes, there's a lot of factors I'm neglecting. IMO, it would be worth spending 40 bucks or so, and installing an outside oil cooler, to learn that the trans. itself is OK.. imp