@Francesco1636 Hi. If it is in his house rules, and you and your friend reserved his home for only 2, yes he can say no unregistered guest in his home. It doesn’t matter how large his home is. And it is a reasonable rule. Also, by airbnb TOS, all guests must be registered for the host to be covered if anything goes wrong.
In your post, you even mention overnight. If it’s just for coffee or dinner, it’s best to meet at a restaurant or a coffeeshop.
I do, however, want to give you kudos for asking the host in advance because so many guests don't and end up getting their stay cancelled. But since the host gave you the option of paying extra, and if it’s that important that your friend or family stay over, why not agree to pay him the extra fee. That’s what I would do as a guest.