I have a hunch that the following is true:
Conjecture. Let $\kappa$ denote a regular cardinal. Let $F$ denote an endofunctor of $\mathbf{Set}$ such that the category of all $\kappa$-small sets $\mathbf{Set}_{<\kappa}$ is closed under $F$. Then $F$ has an initial fixed point of cardinality at most $\kappa$ and terminal fixed point of cardinality at most $2^\kappa$.
My reasoning is roughly that by the usual transfinite composition argument, we should be able to witness the initial fixed point of $F$ as a quotient of a coproduct of at most $\kappa$-many objects of size strictly less than $\kappa$, which should have cardinality at most $\kappa$. Similarly, we should be able to witness the terminal fixed point of $F$ as a subset of a product of at most $\kappa$-many objects of size strictly less than $\kappa$, which should have cardinality at most $2^\kappa$.
However, I'm having trouble with the details of the argument; in particular, how do we use the fact that $\mathbf{Set}_{<\kappa}$ is closed under $F$ to deduce that we need at most $\kappa$-many objects to be added or multiplied together? I included an assumption that $\kappa$ is a regular cardinal in order to help with this, but I'm having trouble using this "helper assumption".