I have taken a look into the waterfall example. This got me an idea to use particles for this. I did some rudimentary testing with some simple quads and it is going into to right direction. But in my case I have to render the animation multiple times in simultaneously, if each uses a few thousand particles that could impact performance really fast.
Another Idea I got is to deform a mesh and make it behave like a fluid. But I have no idea how to do this in Godot or if this is even possible.
Here is an visualisation of what I want to animate. The fluid animation does not need to be the same as in the video, this is just an example.
