You shouldn't need to feed that much in one feeding, even to put weight on. You should be able to feed 3 lbs dry weight of alfalfa, 1-3 lbs dry weight of beet pulp, and then a ration balancer or a simple vitamin supplement. That along with his hay SHOULD be all he needs to put weight on slowly.
And really, he's at a fine weight in those pictures. I would say he only needs hay and a ration balancer, nothing extra (maybe half a 2qt scoop of whole oats or a full scoop of alfalfa pellets, just for a little something extra, but nothing more). I know it sounds contrary, but most horses WILL gain and keep weight on when you take them OFF feeds high in sugar and starch.
You're only feeding 1.5 cups of beet pulp and 1.5 cups of alfalfa a feeding. That is NOTHING for a horse. It would be like you eating half a peanut butter sandwich or a 1 oz snack bag of chips. When you soak both, it should triple in size, approximately, but the dry weight is what matters, not the finished volume. If it's too much volume, don't use as much water, or any water. Horses can eat dry beet pulp just fine, especially the shredded kind.