Megalomaniak
With this I have found the solution to my problem, and it is that I had forgotten to share the view of the Scene nodes and how it was all assembled.
It begins all with the main Node, being the Level Generator, and I looked to the Pause Button's Node Parent and made sure that it's Process Mode was set to Always, and it still didn't work.
Then I realized that everything was the child of a SubViewport that I used to specifically render the scene.

After that I thought of moving the ControlHUD canvas away from the SubViewport Node, and it is now a child of the root LevelGenerator.

Now it all works as intended, even after only setting the specific Pause Button Node's Process Mode to Always and the rest to Inherit.
Apparently as long as the Node is not a part of the SubViewport it will be able to work.
Hopefully this helps out to anyone out there! Thank you once again to Megalomaniak for the prompt that helped sort this out.