No, you are not being over-sensitive.
This careless 'humour' has obviously awakened some distress from your past of which your FIL may well be unaware, but his heavy-handed humour is misplaced when used to the mother of the most beautiful baby in the world - which is any of them!
I think you are entitled to make your displeasure clear. When he says it, say firmly "Don't say that Fred, she doesn't like it ..." and he will no doubt say that she doesn;t understand. You can then reply, "No, but I do, and I'm stopping it becoming a habit before she does understand."
If he throws a strop, more fool him - and he can miss out on seeing his grand-daughter for however long before he sees that he is being insensitive.
Your first responsibility is to your child, then you, then your husband,. and your FIL is way down the list, so it's not fair that he is upsetting you, albeit unintentionally.
Failing that, ask your husband to have a quiet word with his dad and explain that you don't like the 'nickname' he is using, and would he like to think of something else.
Resolve this now - you have enough to cope with without this bothering you when there is no need.