The closest parallel I can find is that of the portraits in the HP books. Some are livelier than others (e.g., Phineas Black), but even they are supposed to be shadows of the person who has passed on. In this sense, I think even the tiniest piece of consciousness would render someone quite lively. Clone!Snape might have less soul than his original self, but he'll probably be just as Snape-ish.
I don't think canon itself fully answers how a Pensieve works. Granted, the storing-soul-in-Pensieve part is completely my own invention, but I've often wondered whether a person remembers a memory after s/he deposits it into the Pensieve. Fandom seems to generally accept that a "shadow" or paler version of the same memory still remains, and if that's the case, maybe Snape has come up with some ways to deposit his soul as well.
No need to feel ridiculous, since you really made me think here, and thinking is always good. :)