Is there another well-pointed elementary topos satisfying internal choice without natural numbers object?

Question

Is there an elementary topos which

1. Is well-pointed
2. Satisfies the internal axiom of choice
3. Does not have a natural numbers object; and
4. Is not the category of finite sets?

Answer 1

Yes: take any non-trivial ultrapower of $\mathbf{FinSet}$. Your properties 1–3 are all expressible in the first-order language of categories, so they are preserved by ultrapowers.