I would not spend one dime on it at this point. Pull the transmission dipstick and examine the fluid. If it’s blackened or resembles sour milk then you can figure the trans is trashed so the radiator is a non-issue at this point.
Considering the coolant leak, there’s a chance the locking up could be the engine either overheating or it has suffered a blown head gasket. Coolant can enter the combustion chamber(s) past a blown gasket and cause what is called “hydrolock”.
To prevent problems in the future, I would advise disconnecting the radiator trans fluid cooler completely and installing an aftermarket fluid cooler.
These can be had cheap, they’re easy to install, and worth their weight in gold. Not only do this eliminate more transmission fluid heat they also get totally rid of that leaking radiator cooler problem.
You can get a fluid cooler for anywhere from 50-75 bucks and that’s dirt cheap insurance is the way I look at it.