Since the question uses semisimplicial sets, it makes sense to point out the following rather elegant model for the classifying space $\def\B{{\sf B}}\B G$ as a semisimplicial set: declare the set of $n$-simplices of $\B G$ to be the set of principal $G$-bundles over the $n$-simplex $Δ^n$ (the same $n$-simplex that is used in the construction of geometric realizations of semisimplicial sets). The face maps restrict a principal $G$-bundle over $Δ^n$ to its faces.
Now given a semisimplicial set $K$ together with a principal $G$-bundle $P→|K|$, the classifying map $|K|→|\B G|$ is the geometric realization of the simplicial classifying map $K→\B G$ that sends an $n$-simplex $σ$ in $K$ to the restriction of the principal $G$-bundle $P→|K|$ to $|σ|⊂|K|$.
As usual, to ensure that $\B G$ is a simplicial set and not merely a simplicial class, any of the standard tricks can be deployed. If the cardinality of $K$ is limited (the question talks specifically about finite $K$), then we can simply require the underlying sets of total spaces of principal $G$-bundles to be subsets of a fixed set, e.g., for a finite (or countable) $K$ we can take a set of cardinality continuum.
Concerning the question about using an open cover and its pairwise intersections to present a principal $G$-bundle, it is true at least in the contexts where good open covers are available, e.g., smooth manifolds. This follows from Proposition 4.13 in Numerable open covers and representability of topological stacks and Lemma 3.3.28 in Equivariant principal infinity-bundles, which show that given an ∞-sheaf of groups on the site of cartesian spaces, its delooping as an ∞-presheaf is in fact an ∞-sheaf, so the descent datum for a principal $G$-bundle can be expressed as a morphism from the Čech nerve of a good open cover to the delooping $\B G$ as an ∞-presheaf (i.e., objectwise).