Flagellum Dei

The 150 points is cumulative, not a single-round score - that's the key thing, and it makes this far easier than it reads. Unlock the mounted archery competition (available after the Horses for Courses quest, at the Nomads' Camp in the Trosky region - speak to the horse handler and the archer), then just keep entering and racking up points across runs until the total clears 150. Practise your horseback bow aim and it'll come in a few attempts.