$\newcommand{\ms}{\mathscr}$Let $\ms C$ be a compact class. Let $\ms C_u=\left\{\bigcup\ms F:\ms F\subseteq\ms C\text{ is finite}\right\}\;,$ and suppose that $\{F_n:n\in\Bbb N\}\subseteq\ms C_u$ is centred (i.e., finite intersections are non-empty); we want to show that $\bigcap_{n\in\Bbb N}F_n\ne\varnothing$.
Let $\displaystyle\Phi=\prod_{n\in\Bbb N}\ms F_n$; a point $\varphi\in\Phi$ is a sequence $\varphi=\langle C_n^\varphi:n\in\Bbb N\rangle$ such that $C_n^\varphi\in\ms F_n$ for each $n\in\Bbb N$. Then
$\bigcap_{n\in\Bbb N}F_n=\bigcap_{n\in\Bbb N}\bigcup\ms F_n=\bigcup_{\varphi\in\Phi}\bigcap_{n\in\Bbb N}C_n^\varphi\;.$
(This is just distributivity of intersection over union, though you may have to think about it a bit if you’ve not seen an instance of the infinite distributive law before.)
Thus, $\bigcap_{n\in\Bbb N}F_n\ne\varnothing$ iff there is a $\varphi\in\Phi$ such that $\bigcap_{n\in\Bbb N}C_n^\varphi\ne\varnothing$. The existence of such a $\varphi$ will follow from König’s lemma after we build a suitable tree.
For each $n\in\Bbb N$ and $\varphi\in\Phi$ let $\varphi_n=\varphi\upharpoonright\{k\in\Bbb N:k. Viewed as a set of ordered pairs, $\varphi=\bigcup_{n\in\Bbb N}\varphi_n$; if $\varphi$ is viewed as a sequence, $\varphi_n$ is its initial segment of length $n$. Note that $\varphi_n\subsetneqq\varphi_{n+1}$ for each $\varphi\in\Phi$ and $n\in\Bbb N$. Let $T=\left\{\varphi_n:\varphi\in\Phi,~n\in\Bbb N,\text{ and }\bigcap_{k
$\langle T,\subseteq\rangle$ is a tree. The root of $T$ is $\varnothing=\varphi_0$ for all $\varphi\in\Phi$. The predecessors of $\varphi^n\in T$ are the $\varphi^k$ with $k. Each $\varphi_n\in T$ has at most $|\ms F_{n+1}|$ children, so $T$ is finitely branching. Finally, for each $n\in\Bbb N$ we know that $\bigcap_{k\le n}F_k\ne\varnothing$, so there must be some $\varphi^n\in T$, and $T$ is therefore infinite. By König’s lemma there must be a $\varphi\in\Phi$ such that $\varphi^n\in T$ for all $n\in\Bbb N$, and therefore
$\bigcap_{n\in\Bbb N}F_n\supseteq\bigcap_{n\in\Bbb N}C_n^\varphi\ne\varnothing\;,$
since $\ms C$ is a compact class.
Closing $\ms C_u$ under countable intersections shouldn’t be a problem.