We have found one software suitable in our database for this conversion.
It looks that both .bmp and .bm files are in most cases raster images, so nothing in theory should prevent a potential bmp to bm conversion. You just need to use a converter that supports both of these files types, for example the popular XnView MP.
Updated: October 13, 2016
A free (for non-commercial purposes) powerful cross-platform media browser, viewer and converter. Compatible with more than 500 formats and also supports export to 70 different file formats. It works on Windows, Linux and macOS (OS X).