In no signal, I have some embedded windows that the user can open to take notes and view the map. However, if these windows are visible, these windows always render ontop of my pause menu.

After some searching online, I found godotengine/godot#99970, which mentions a line in the Godot documentation for CanvasLayer:

Note: Embedded Windows are placed on layer 1024. CanvasItems on layers 1025 and higher appear in front of embedded windows. https://docs.godotengine.org/en/4.3/classes/class_canvaslayer.html

Unfortunately, you cannot set this layer index in the inspector, so you have to use a script to set the layer instead. Changing my pause menu to layer 1025 fixes the issue.