The only way to produce Sothink .lmk logos from BMP pictures would be to use the original software that makes these files - the Softhink Logo Maker. It is unlikely any other program is capable of making .lmk logos, so Softhink Logo Maker is your only option.
Convert Standard Windows bitmap image to Sothink Logo Maker logo.
We have found one software suitable in our database for this conversion.
Microsoft Windows software - convert bmp to lmk on Windows
A logo design editor
Convert from bmp file format
The table with program actions contains information about what each program is capable of doing with their files. This may be a good pointer for further searches for certain file conversions and help you find the converter you are looking for if our search result is insufficient for you. However, it is far from perfect and may sometimes show results which are not really usable because of the nature of the software handles the files. So a conversion that appears as possible, just because of matched actions, may in fact not be possible.