I have a HDA that renders an image after pressing a button on the HDA. I'd like this image to be imported as a sprite and referenced by a scriptable object. I'd like this to be automatic, so when the image is rendered I can run a script in Unity to setup the references in my scriptable object.

Is there functionality to have this callback when the HDA has finished rendering?

I know you can have callbacks after the HDA has finished cooking. Is there a method of having a callback once a single python module has been executed?

Nope, script callbacks are only done at end of cook.