I think one thing that might be going on here is that, the set of animal names that one might use to refer to someone endearingly is different between Hindi and English.
Growing up, my parents and aunts and uncles would call me and my cousins things like chuuhaa, billii, bandar, makkhii, chamgiidaR, and probably some other animal names, but all of them in an affectionate way. Sometimes it had to do with one of us exhibiting an attribute that is also exhibited by the relevant animal. For instance, I often woke up very late looking disheveled, which led to me getting called chamgiidaR. Or, when I was being very quiet during family gatherings, I would get called chuuhaa. But sometimes, animal names were just a general term of endearment without any real connection to an attribute that one of us was displaying. One of my cousins got called billii frequently, and I can really think of any distinctively cat-like attributes she displays (and in particular, her eyes are just brown and not very cat-like). As far as I could tell, it was just a way of expressing affection.
In an English context, I've seen my anglophone relatives call their children "monkey" and "silly goose" in an endearing way. I can at least kind of imagine "mouse" being used endearingly, though I'm not sure I've actually witnessed it. But... "Cat"? "Fly"? "Bat"? It's sort of a strain on my imagination to picture those used as terms of affection in English.
My point is, maybe you won't be able to find a word that you can consistently use as a good translation of a word like billo. If you can identify a specific cat-like attribute exhibited by the person to whom the term is applied, it probably makes sense to use an appropriate word for that attribute in particular --- eg, if the term is being used in reference to eye color, something like "blue-eyed" or "clear-eyed" or whatever could be a good translation, as has been suggested above. Maybe even "cat-eyed," if they have particularly catty eyes. But if there's no obvious cat-like attribute being exhibited, it may just be a general term of endearment, and maybe the best thing to do is to translate into English using a word that actually gets used as a term of endearment in English that's appropriate for the context (eg, "pumpkin" or "sweetheart" or "snookums" or whatever).
