Why is it that compressing a file already compressed by LZH can't compress it further?
Like if you gzip a .gz file you don't get any savings.
It seems like you could use a pseudorandom but reproducible (by using the same seed) perturbation.
Then compress the perturbed file, and repeat.
To decompress, perform LZH decompression and then reverse the perturbation.

This is an obvious strategy, so there must be some reason why it hasn't already been implemented.