How can we decompose any image (e.g. medical image, satellite image) into high- frequency and low-frequency parts by using a bilateral filter?
Since the Bilateral Filter is Spatial Invariant and Non Linear (The output is linear combination of the data, yet the weights depends on the data in Non Linear fashion) it can't be formulated as a Spatially Invariant Filter / Convolution.
So the Low Pass / High Pass concept in it classic form doesn't hold here.
Yet you can still employ the Image Pyramid concept here with your operator being the Bilateral filter and everything will work perfectly (Namely perfect reconstruction).
It is actually very effective.
Basically, no. The bilateral filter is nonlinear. It is meant to preserve edges (local, high-frequency), while reducing noise (global, broadband). Hence, as low/high frequency separation is a concept akin to Fourier, hence quite related to linear filters, your question is unlikely to get a positive answer, without further details on what you call low/high frequency.