I really wouldn't worry too much. Does your DS enjoy finger foods at all? I know Lilah really only wanted to self-feed once she could do it, and always ate better if I gave her finger foods.
You're right that rice milk is very different from cow's milk in that the protein content is very low. I'd give about the same amount as you would alll other milks, though, and just concentrate on getting other sources of protein in there. Any reason you're using rice milk instead of soy? Just curious. Soy milk is more like cow's milk in its protein/carbohydrate balance, but lots of people worry about soy allergy.
And FWIW, the "recommended" amount of cow's milk is 16 oz. Too much can actually cause anemia in young children (here is a big benefit of using non-cow's milks and also of extended nursing). Lilah never took that much milk, but she still nursed 2X a day until she was 22 months, and I think she got plenty that way.