Depending on how the horn is, it could hurt, sure.
But those horns seem reminiscent of antlers, and for deer antler's, it actually isnt painful at all, once fully grown.
and i'm inclined to believe that it doesnt, because despite that author including that dumb anecdotal comparison, no, in real life, even professionally, fully trained service animals, will yelp in pain if they are hurt.
.
But even on the chance it does hurt them, he could probably take a page out of a cow farmer's book, and take care of it when they are young or something, when the pain is negligible, and keep the pain to a minimum.