i think if you attend to it promptly you'll get away with just a fluid change. a tranny flush would be good, if you can find a place that isn't a bunch of hammerheads.
it's entirely possible that tranny pressure is higher than coolant pressure at that point, and the leak was one way.
The trans "cooler" is rarely much more than a metal line run through the bottom tank; it's not soldered sheetmetal like the water side of the radiator. Almost certainly, the trans leak was a corrosion pinhole, vibration or other crack, etc, and is fairly small; trans fluid was probably pushing into the radiator before it blew. I bet nearly no radiator juice got into the trans.
I know you know the test -- run it hot, put in NEUTRAL (not park), check the fluid... I know you're cringing from doing the test, I would too!