Do be careful with feeding dry beetpulp- yes they do generally like it, however, once it gets into the stomach, it swells there rather than swelling in the bucket with the hot water you have added. If you give them too much dry, it will swell too much and give them a tummyache (or worse).
We use Kaolin and Morphine mix when a goat is actively scouring- but it sounds just like yours is loose due to the nice rich grass. It wouldn't impact the milk though if you do need to use it. Just be cautious trying to buy it- becuase its got morphine in the chemists don't really like selling it....
I would be trying some other carbohydrates for her- flaked barley, flaked peas, maybe some bran to try a bran mash or Alflafa.
Beth