Orthography is not intuitive. To test my native speaker instinct, I'll pick a case that is. Imagine a user whose name was "Praise_Him". To me, it would be more natural to say "Praise_Him's post" than "Praise_His post"; the former might give me a second's pause, but the latter would make me reread the sentence. Thus, at least the way I use the language, a proper name which incorporates a pronoun is possessivized as a whole, and cousin_it's is correct. But "Its" and "It's" are homophonous, so it wouldn't matter to me much.