BC = BC1,2,3,4,5,6H,7 = suite of lossy texture formats used by PC GPUs. All these need to support efficient random access and work by encoding 4x4 blocks of pixels to a fixed-size format (either 64 or 128 bits per block). They are decoded as part of the texture fetch. It is safe to assume that at any point in time, the vast majority of texture data in GPU memory is in one of these formats.
BC1-7 encoding is lossy. BCPack is a lossless coder on top of the lossy BCn data. As such, comparing to Kraken is fair.
Both are run on reading data from disk into memory, thus they don't have the random-access requirement that textures in memory do, and indeed better entropy coding is a likely thing to try. I can't comment on what exactly BCPack does or how its performance compares to Kraken, either "full fat" or the PS5 subset; this is covered by NDAs with both MS and Sony.
For BC1-5 there are several well-known, easy lossless transforms that tend to significantly increase compression ratio (5-15% reduction is common, depending on the BCn format and the data) with your usual LZ. This is really simple stuff. For example, a 64-bit BC1 block is 32 bits of color endpoints then 32 bits of 2-bit indices for every pixel in the 4x4 block. Reordering data to separate endpoints from indices, putting them into separate blocks (and so they get separate Huffman tables in say a Deflate stream), helps massively for coders that don't use the low-order bits of the position in the stream as context. This is a lot better than straight Deflate/Kraken on this type of data and both the Xbox Series S/X and the PS5 support it as part of their output write from the decompressor to memory. (Not quite free, but very nearly so.)
BCPack is more sophisticated than that, PS5 decided that the basic reordering plus Kraken was good enough.
BC6H and BC7 have many modes and a more irregular block layout and such trivial transforms don't work. Oodle Texture has "BC7Prep" which is a lossless transform that can be run on BC7 blocks to make them more amenable to compression by byte-aligned LZs, and is easy (and very fast, often >180GB/s) to undo on the GPU. It mostly boils down to making the transform aware of the different modes. BC6H we don't have anything in particular yet because it was <1% of the texture data sets for all games we looked at so there were better things to spend our time on.
In short, XBox Series S/X have regular Deflate and BCPack which is a lossless coder for _only_ BC1-7 data (itself lossy), PS5 has Kraken, both support certain simple on-the-fly transforms on BC1-5 blocks as part of the decompression process, and both can use Oodle Texture BC7Prep via GPU compute shader for BC7 where the simple transforms don't work.
BCPack is aware of BC1-7 data, Kraken is not, and both can potentially get much better results when the BC1-7 encoder feeds them the kind of redundancy they know to exploit.