We further the classification of rational surface singularities. Suppose (S, n, k) is a 3-dimensional strictly Henselian regular local ring of mixed characteristic (0, p > 5). We classify functions f for which S/(f) has an isolated rational singularity at the maximal ideal n. The classification of such functions are used to show that if (R, m, k) is an excellent, strictly Henselian, Gorenstein rational singularity of dimension 2 and mixed characteristic (0, p > 5), then there exists a split finite cover of Spec(R) by a regular scheme. We give an application of our result to the study of 2-dimensional BCM-regular singularities in mixed characteristic.