If the file is an illustrator (vector) file, reduce it's size to 32px high and alter the color so everything is either black or white (no grays).
Save the file as an Illustrator file once you have done the above.
Then open the Illustrator file with Photoshop and rasterize it. Note, simply opening it should rasterize it, if by chance it opens as a smart object, then merge it with a blank layer to rasterize it or use the contextual menu to rasterize the smart object layer.
This should result in an RGB 33x32px image. (or roughly that size)
Image > Mode > Greyscale to remove the color data from the image.
Image > Canvas Size to set the canvas to 128px wide.
Image > Mode > Bitmap to convert the file to bmp format. If you want the size of the image to remain the same, then you need the output to match the input - so 72PPi -- this is a VERY low resolution for a bmp image. This isn't going to reproduce wonderfully in print. But in print you'd probably not use a BMP image anyway. You have not described why you need the BMP format.
The result is a black circle with some blob in the middle, see below (minus the pink).
(Edges could be a tad smoother, but that is going to require you to manually use a small brush, like 1px in diameter, and go in and add or remove individual pixels to better define areas. The BMP format does not allow "grey" pixels so you won't get any anti-aliasing.)
If your desire is a clearly identifiable one color logo, the logo needs to be reworked to meet this size. It is simply not a good logo for one color at this size.