Assuming $MA_{\aleph_1}$, the answer is positive.
Let $P$ be the collection of all positive measure finite intersections of elements of $\mathcal{K}$, ordered by inclusion. Then $P$ is a ccc uncountable partial order so (by $MA_{\aleph_1}$) it contains an uncountable centered subset $Q \subseteq P$. If we let $\mathcal{A}$ be the collection of all elements of $\mathcal{K}$ that contain some element of $Q$, it follows that $\mathcal{A}$ has the finite intersection property and therefore $\bigcap \mathcal{A} \neq \emptyset$.