What Is the Bilateral Filter Category: LPF, HPF, BPF or BSF?
Neither. To me, filter classes using the notion of frequency bands (low-pass, high-pass, etc.) can be used  safely in the linear case. And the bilateral filter is nonlinear. Edges are not really high-frequency: they often have sharp variations across the edge, but slow variation along it. Hence linear directional filters often have a derivative part and and orthogonal smoothing part.
I would consider the bilateral filter as an edge-preserving smoother, a broad and somewhat imprecise class.
Suggested reading: Fast and Provably Accurate Bilateral Filtering, 2016

The bilateral filter is a non-linear filter that uses a range
filter along with a spatial filter to perform edge-preserving
smoothing of images. A direct computation of the bilateral filter
requires O(S) operations per pixel, where S is the size of the support
of the spatial filter. In this paper, we present a fast and provably
accurate algorithm for approximating the bilateral filter when the
range kernel is Gaussian. In particular, for box and Gaussian spatial
filters, the proposed algorithm can cut down the complexity to O(1)
per pixel for any arbitrary S. The algorithm has a simple
implementation involving N+1 spatial filterings, where N is the
approximation order. We give a detailed analysis of the filtering
accuracy that can be achieved by the proposed approximation in
relation to the target bilateral filter. This allows us to estimate
the order N required to obtain a given accuracy. We also present
comprehensive numerical results to demonstrate that the proposed
algorithm is competitive with the state-of-the-art methods in terms of
speed and accuracy.