Let $k$ be a field of characteristic $0$. In this paper we describe a classification of smooth log K3 surfaces $X$ over $k$ whose geometric Picard group is trivial and which can be compactified into del Pezzo surfaces. We show that such an $X$ can always be compactified into a del Pezzo surface of degree $5$, with a compactifying divisor $D$ being a cycle of five $(-1)$-curves, and that $X$ is completely determined by the action of the absolute Galois group of $k$ on the dual graph of $D$. When $k=\mathbb{Q}$ and the Galois action is trivial, we prove that for any integral model $\mathcal{X}/\mathbb{Z}$ of $X$, the set of integral points $\mathcal{X}(\mathbb{Z})$ is not Zariski dense. We also show that the Brauer-Manin obstruction is not the only obstruction for the integral Hasse principle on such log K3 surfaces, even when their compactification is "split".