I'm a language pedant, but I'm happy with "I've got a little list".
I agree that "I've got" (I have got) is not a good replacement for "I have", but it would be suitable for "I have been to fetch", or (at a push) "I have gotten together".
If the phrase were to be altered, I would prefer "I've made a little list". This would cover either of the alternate meanings given above, and clarify that this is a list that Koko himself has put together.
But as we don't know W S Gilbert's precise intended meaning, we need to stick to the words as written, even when modern productions change the contents of the list at will.

(My opinions only)