Which Approach Is Better for Decomposing an Image into High Frequency and Low Frequency Components?
The bilateral filter is just that: a filter. It does not decompose the image into anything. It just attenuates "medium" frequency content in a clever manner. In other words, it smooths things that are mostly smooth further, and leaves sharp edges sharp. What you are left with has more low and high frequency content (on a relative basis) than what you started with.

The Fourier transform, on the other hand, actually does reveal the entire spectrum for an image. Unlike a filter, it does not change the information content of an image. It just rearranges it into the frequency, rather than spatial, domain. The bilateral filter does not reveal the spectral content of an image, so really only the Fourier transform (or one of its related transforms, like cosine and wavelet) can be used to get the high and low frequency components of an image.