Another update:
So, EXR Works great. Except for when your other software doesn't support it and you have to do file conversion. In my case, TerreSculptor (terrain generation software) doesn't support EXR so I have to batch convert thousands of files.
Of course, tools exist to do this and it would be fine for a few files here and there. But in my case, dealing with 16384 files, many tools will outright fail due to bad program design and some will be very slow because it doesn't utilize multiple cores.
Affinity Photo does work, but the UI hangs and sometimes crashes, so it's not suitable.
The popular command line ImageMagick tool only has EXR support when compiled from source with extra OpenEXR libraries, something that many will struggle to do and is cumbersome (I have yet to attempt this).
I found a script for blender that essentially batch converts any image in a source directory and saves to EXR (since blender has EXR support built in). But the script is for blender 2.79, but fine, I tried it and it basically doesn't work. The resulting images are all gray somehow.
So, I'm basically stuck at this point until Godot gets support for 16 bit PNG (the author of TerreSculptor said that it will be unlikely for him to add exr support in the near future, due to lack of funding for the project).