About Cathedral Windows
 

I have heard that it does not need batting because of the way it's made, but if I did one, with batting, would it then be TOO thick?

I personally would not add batting because CW is already a heavy quilt from all the fabric layers. However, people have done it with batting. Here is one link:
http://thejiltedballerina.blogspot.c...-tutorial.html

Edit: If adding batting, I would definitely use a thin batting.

Some people do put a bit of batting under the window. This blog is an example - http://hyena-in-petticoats.blogspot....-tutorial.html - and another - http://thejiltedballerina.blogspot.c...-tutorial.html

I would try it both ways and decide which one you like best.