Good news Beet greens tops is ok for buns.    For more info on different safe and not safe foods for bun you can check out

houserabbit.org      click care on the top left circle     then FAQ diet   and you will get some lists.

Glad  you found some good uses for beet greens.  And you have a horse how cool

Fresh Beet pulp – would that have part of the root? I am so not a beet fan so I steer clear from beets – sorry for my lack of beet knowledge.. But Beet green tops are fine and the beet root fits under the same category as broccoli, brussel sprouts, cabbage. Some rabbits may be sensitive to beet root, so you just want to watch carefully, feed a little bit when introducing. I think though of pulp as the stringy and/or harder stuff to digest that is left behind. If that is the case then I am not really sure. It may be better to go with just fresh cut beet pieces or better yet, just the tops.

Maybe someone else will know about the “pulp” aspect. 

I googled beet pulp and Wikipedia had a good description, though only for horses. Since you said you already soaked it, it is not harmful to horses.
I checked the HRS site and only found one mention of beet pulp in the content of a pellet, listed under roughage. There were several mentions of beet greens as recommended however.

I’d opt to be safe and give him beet greens, and see if that satisfies his craving for the beet pulp as a substitute.

I think, and correct me if I’m wrong, but that beet pulp for horses is mostly a filler?

Rabbits and horses have a TON in common (stasis literally equals colic in all ways-causes (mulitple including idiopathic) and treatment, the darned cecum etc) so I can see why you’d want to do beet pulp. Rabbits are a lot more sensitve to carbs and sugar though, they don’t get worked like a horse does (most horses are technically atheletes) so even if safe, could really end up being stored fat