Does anyone have a handy way to clean corroded batteries out of a toy? I can't find anything that seems to get all the gunk out.

After cleaning it all out with a cloth you then need something metal like a wire brush, I have a tungsten tipped point that slowly scrapes it all off but the last remnants can also be removed with sandpaper as well. It's usually possible to get rid of it all if you're patient.

Make a paste of baking soda for the terminals. It will neutralize the batteries acid. Flush with water (if possible) and let air dry.
It's the same cure for corroded automobile batteries.