I think you got it figured out already - a shader with several layers mixed or even just animated UVs would do the trick. I can only add that manually distorting UVs can imitate local turbulence / water flowing around rocks.
So this is something I've had a hard time figuring out for a good while now. it seems like something that was avoided in alot of games in the PS2/Xbox/GC era, but a few games and some mods from the era seemed to try and solve it. Does anyone here have any content for doing running water effects within those specs? I'm…