Can sprite characters in a 3D game sit inside transparent water?

I have a 3D game with 2D sprites for characters (I know that might be a bit odd but I really like the style, been doing it for a while). I'm trying to put a flowing river into the world and have run into a bunch of problems which I've shown in the screenshots below because I didn't know how to describe them...

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The left-most screenshot shows how I want the dog character sitting in the river except I want the river to be transparent and animated.

My first plan was to have an animated sprite turned 90 degrees (as shown in the two right-most screenshots) but I understand there are problems here as both the river and the dog are UI-type objects correct? Is there some way around this that I'm missing?

I've considered that maybe I need to explore and learn how to make animated textures and put that on a mesh but as seen in the second screenshot, there's a weird issue where the dog no longer "sits in" the water when the shader is transparent.

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So I'm just a little lost and unsure which way to go about this. Would really appreciate some help :)

(Edit: I refuse to use Unity's or whoever else's pre-made water! I just really want to make everything my self you know?)