I don't know much, if anything, about heads-up play, but I would probably give up on the river. As you said, hero blocks all the missed draws and none of the board. I believe villain can bluff-catch with a large portion of their range. The only justification I can see to bet the river is that it seems villain is a bit of a nit, as they didn't check-raise the turn or river, and they don't block any of the missed draws.