Oh, I totally see this kind of connection between Sirius and Snape. Before we knew Snape was a half-blood, I used to have him being held up by Mrs. Black for Sirius's edification = "See what a loyal pureblood, a true Slytherin is like, and see what I got stuck with as a son" - and that Regulus and Snape would have bonded perhaps, as targets of Sirius's venom, as well as fellow-feeling on the whole pureblood issue.

Snape as half-blood is more problematic, but I do think Sirius *would* have played the Black family card over him, knowing that Snape desperately wants what Sirius has and repudiates (at least, the racist ideology of it - I don't think Sirius ever quite repudiates the influence of his name, and his breeding is very clear in his sense of noblesse oblige toward the Order and his whole bearing. I think Sirius, despite hating his family, very much would have traded on his name to get what he wanted), and thinking less of him for it, while Snape hates him for having it and not wanting it.

Which is kind of what you already said, so yes, I agree. *g*