If the heat exchanger(trans cooler) in the radiator is leaking then you have to change the whole radiator.
And best to flush engine and trans, and then refill, IF this is the problem
You could unhook both trans cooler lines from radiator
Then use a Cooling system pressure tester(hand pump with pressure gauge on rad cap fitting) to pressurize the radiator and see if it is leaking pressure, and if so you would see coolant start to drip out of trans cooler line fittings on radiator.
You can drain the ATF into a pan then pour off the top ATF and put the rest in a clear glass container, coolant/water would settle to the bottom
But if trans is cold and it happens then I would look elsewhere
Check Vent tube and trans cooler tubes, could be a kink in the metal tube blocking flow, or obstruction
I have previously put air to the cooler lines and have good flow in all directions.
Curiosity; First thought regarding a boil one would think great heat, yet I put my thumb over the filler tube to restrict back flow without concern towards being burned all be it the “trans fluid” exiting the filler tube was barely warm,,, is it reasonable to think were it the liquid being under the trans fluid that is doing the expanding, and hence forcing the upper floating layer of trans fluid out in precession?
In so far as the test you mention by way of removing cooler lines and pressurizing the rad, in the absence of a tool with gauge would it not also work to remove those lines as you say and bring motor to full operating temp with rad cap secure and inspect for leakage of “coolant” at the trans cooler line connection points of the rad, or would that degree of pressure not be insufficient (and too the cooler lines would leak if not purposely blocked)? I am thinking if it is operating temp pressure in the rad allowing coolant to enter the trans then this would show consistent and still leak as described? If to do it this way should I be concerned about the trans overheating and being damaged?
I was thinking in the event of this in fact being the problem then, pull the cooler lines and stop up the connection points on the rad itself, then install a standalone cooler for a fraction of the cost of either a new or re-core rad???
I shall be crawling on the floor and know space is very restricted between the floor pan and the trans, so may I please ask where do I feel for the vent, or am I looking for an actual tube of what material?
After thought, could one drain the trans by running motor with cooler lines disconnected and save dropping the pan?