The closer you are to the root, the better your chance it will pull out instead of break off. Also, if his mane is very dry or brittle it will break. Also, the smaller the section, the easier it will be to pull without breaking. Maybe try a different comb. I have seen pulling combs have a bit of a sharp edge on the sometimes. That particular comb looks to have that sharper edge between the tines.
ETA, if you pull it after he's warm from a workout, it releases easier, but don't do it if the hair is wet, that will break it for sure.
"Keep a leg on each side and your mind in the middle"