I don't like to feed beet pulp to horses, but horses have choked on the dry pulp, so please soak it overnight if you decide to feed it! (I know that you can find vets who deny that the dry has caused choke, but they're wrong! Easy to imagine, since it's just dusty, totally dry fiber!)
To be fair, Northern, I think it is important to recognize that horses can choke on ANY feed. Horses can choke on hay/grass (less likely, but it can/does happen), pelleted feeds, etc.
As with any feed change, gradual change is in order. Start small, as suggested, and work up. Also, know that some horses may not take to bp right off. I like to use bp as one component of a feeding plan -- at the moment our's consists of bp, alf. Pellets and rice bran.
Rather than just approaching it as adding the bp, though, do see if you can find out what grain she is using at the moment as a change there may help to bring about a more effective/healthy weight gain. The quality of the hay is also important. Having hay at all times is a great step, but if the quality of that hay is not what it could be the horse isn't getting all that it could from having it there.