You leave a lot of questions unanswered. What architecture of microprocessor is this? Is everything in one binary firmware blob, or is there storage with a filesystem?
Most likely the image data is in a raw format embedded in the firmware binary. For a small embedded system, I would be surprised to find a "BM" header. What I would do, is disassemble the firmware, and look for references to large "blobs" of data. They could be drawing smaller sprites, or it could be copying a large image (you haven't really specified.)
The details about the LCD will be helpful here also. Most likely a datasheet will tell you what format the LCD expects its data in (how many bits/pixel, etc.) If you can figure out (electrically) how the LCD is connected, then you can possibly figure out what I/O ports the firmware will access to write to the LCD. Then you can work backwards to see how it is transforming the image data (if at all) before writing it to the LCD.