That may depend on how solidly he goes by Ben, assuming that's his real first name. I go by both my nickname and my full name pretty interchangeably, with a slight preference that strangers/acquaintances use my full name. I'd never sign something as my nickname, wouldn't even occur to me.
Meanwhile, my brother's name is Will and the few occasions I've been around when he needed to sign his full name, he had to be reminded that they wanted William. This could be generational though, we are younger.