Will a grayscale .bmp placed in InDesign print okay?

I would use grayscale TIFF for those images. (Just to clarify, the .BMP format is not the same as the Bitmap mode in Photoshop.)

The advantage of doing a duotone in Photoshop is that you can edit the color application curves before converting to CMYK.

Do not use the BMP format; choose any other image format instead (except EPS, of course).

Their brand book suggests taking a grayscale image and then doing mode> bitmap and using dither,

Hi ​@katielady1 , Also keep in mind if you convert to a dithered bitmap and apply a color (something other than black and white) there will have to be a halftone or stochastic screen applied to the colored dither at print output, which will likely cause an unexpected moire pattern. Make sure you get a hi res proof before going to the print run.

https://en.wikipedia.org/wiki/Moir%C3%A9_pattern

Using BMP grayscale to create a faux duotone in Indesign should not cause a print issue. You would have the same result using a TIFF grayscale. But the description from the brand book using a bitmap TIFF that is dithered will not produce the same result as either grayscale BMP or grayscale TIFF.

Using a bitmap for solid art is OK as long as the resolution is maintained or increased when creating the bitmap.

If the desired effect is a mezzotint, using Diffusion Dither converted to bitmap is OK and will print fine, but the display may be poor due to zoom, and other display settings.

Offhand I can’t think of a reason why it wouldn’t print OK, but I’m curious why you choose to save as .bmp rather than .psd or even .tif