It makes sense that he's still considered a Greystone as opposed to a Fortemp. His father might recognize him as his son, but he's not been officially named a Fortemp due to Ishgardian politics, most likely. We know just how rigid Ishgard is with tradition, recognizing and having a bastard live with the head household is scandal. I'd imagine it'll be an even greater scandal if they made him into a Fortemp. With inheritance and all.
And knowing Haurchefant and his father, it would not surprise me at all if his father planned on making it official, but Haurchefant refused.