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 $\bigcap \mathcal{A} \neq \emptyset$.

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 $\bigcap \mathcal{A} \neq \emptyset$.

Edit: A subset $Q$ of a poset $P$ is called centered if any finite $F \subseteq Q$ has a lower bound in $P$. It was proved by Velickovic and Todorcevic in "Martin's axiom an partitions" (1987) that $MA_{\aleph_1}$ is equivalent to the statement that every ccc uncountable partial order contains an uncountable centered subset.