In order to achieve a prop with water as part of it.. how would you go about doing it?

My first thought was a no-collision surface with a scrolling texture, but it seems like there'd be more to it than that.

no, there sadly isnt more than that, you are correct.
ou could howveer experiment a bit with multiple surfaces and different scrollingspeed/directions, maybe that can make it look more realistic

So I'm assuming the "Render Water" render type does nothing?

I think I tried it once, crashed the game nicely.

Depending on how big the prop is will determine how much of an effect you can do.
Just thinking of the water effect on Kamino it looks wave like.

A good example of a prop with "water" is nab2_prop_fountain.msh. Open it up in Unwrap3d and a hex editor to see the setup involved.