Let $X\neq \emptyset$ be a set and let ${\cal S} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say $C\subseteq X$ is a *choice set* for ${\cal S}$ if for all $s\in S$ we have $|s\cap C| = 1$. As [Bjørn Kjos-Hanssen][1] pointed out, [choice sets do not always exist][2].

Is there an infinite cardinal $\kappa$ and ${\cal S} \subseteq {\cal P}(\kappa)\setminus\{\emptyset\}$ such that

1. for all $x\in\kappa$ we have $|\{s\in {\cal S}: x\in s\}| = \kappa$, every member of ${\cal S}$ has cardinality $\kappa$, and
2. there is a choice set $C\subseteq \kappa$ for ${\cal S}$ 

?
[1]: https://mathoverflow.net/users/4600/bj%c3%b8rn-kjos-hanssen
[2]: https://mathoverflow.net/a/285451/8628