I feed it at the evening feeding(once a day). I started like you did, slowly this past fall. It's good stuff and gets a little extra water in them in winter time. They seem satisfied/eat it right up and look good with good body cover and helps mine keep a trim tummy. My yearling, I watched closely not to over do the pulp as the added fiber could make them too loose. So it's kinda something you have to watch in each indivual horse.....just like people they have their own constitution.
I do about a 1 and 1/2 cups for my yearling and 2 cups for my mature mare, once a day. I will be adjusting this some during show season.
What I do is measure out my dry pulp shreads....put plenty of warm water to cover it...let it soak about 20 to 30 minutes, drain off most of the juice but measure about 1/2 cup of juice on each of their pellets and mix their beet pulp in with it. I try not to over do giving mine that molases juice water as it's sugary but a little doesn't seem to hurt.
Edited to add: the beet pulp is fed in addition to their complete pellets and good quality hay, fresh water.