Hey Kevin,
Run this macro when that window is frontmost in your app. (It may take a little while to run, since there are many elements in the window.)
Front Window Analysis Tool for System Events
TextEdit will open with a bunch of fairly indecipherable text, but what we're looking for is items that start with this text:
«class chbx»
If System Events can see them then there are probably ways to manipulate them directly.
-Chris