Upon actually implementing this shader into my game, I realized that my UI was rendering on both the PrimaryCamera as well as the camera which renders to the window, so I made it so only the PrimaryCamera renders UI now. I plan to make this customizable in the future incase you do want UI to render over the post process instead of under it.
I also added some basic scanlines to the shader which I think really goes a long way in improving the aesthetic. Thanks to sixfold from the Bevy Discord for this suggestion, I may implement more of their suggestions down the line as well.