Assuming this is a raster image with sufficient resolution, I would open in Photoshop, invert to make a positive image, change mode to bitmap. Then place new logo into Indesign, and colorize as needed.
bitmap mode seems to be greyed out, unselectable.
Do the blend modes in InDesign not work for print?
Sounds like image is RGB, change to grayscale first then to bitmap. Note: you can leave image as a grayscale, this can be colorized in ID as well, if you don't need to knockout a background.
Blend modes can achieve close appoximations of desired effect, but as you can see with your example, did not work to make a perfect reverse logo.
I've tried Googling and reading up about the Blend modes, it says something about changing the option in Edit > Transparency Blend Space
When you use Screen for the blend mode your black background has to be absolute black for to be completely transparent. If you want to keep your Transparency Blend Space as CMYK and your logo is a CMYK object, the black background has to be absolute CMYK black (100|100|100|100 and not 0|0|0|100):
Can client supply a vector logo?