Those numbers in the info panel are recalculations based on your working spaces - what you'd get if you converted. Same with the color picker.
Just look in the Channels panel to confirm. If you have a single channel grayscale file in Photoshop, or a CMYK file with the image in the K channel and CMY empty, you'll be fine.
If this is a simple black and white/transparent design, the profile is obviously not that critical - but if it's going to be overprinted, you should be careful to not exceed the maximum ink limit.
For an image or similar, however, you need to use the CMYK profile for the actual printing process, which you should get from the printer. Don't assume anything here, and don't rely on Photoshop defaults. You have to ask.