Set Basis Problem Variant reference for NP-Completeness?
The problem is in fact solvable in polynomial time.

Let 
$$C_0=\Bigl\{c\in C:c\supsetneq\bigcup_{\substack{c'\in C\\c'\subsetneq c}}c'\Bigr\}.$$

On the one hand, if $B\subseteq C$ has the property in the question, then $C_0\subseteq B$: since $c\in C_0$ cannot be written as the union of its proper subsets from $C$, a fortiori it cannot be written as a union of its proper subsets in $B$, hence the only possibility is $c\in B$.

On the other hand, $B=C_0$ does have the required property. We prove by induction on $|c|$ that any $c\in C$ is a union of a subcollection of $C_0$: if $c\in C_0$, this holds trivially; otherwise $c$ is the union of its proper subsets in $C$, which in turn can be written as unions of subcollections of $C_0$ by the induction hypothesis.

Thus, the problem is equivalent to $|C_0|\le K$, which can be checked in polynomial time.