Drizz't the cat
(sounds a bit too much like Fritz the Cat!!)
Oh, I don't know - I generally think of him as Drizzle anyway. (Why no, I don't much like those books. Why do you ask?But, in the inevitable modification of cat names as discussed in this thread, you'd eventually end up calling him "Drizzie".
And then you'd start thinking of Drizz't the drow as "Drizzie". And that'd just get weird.
On the other hand, if you wanted to stick to the Drizz't theme, you could call him Guenhwyvar.