It's important for what I'm trying to do that only the cyan cartridge is used.
The drivers for composite inkjet printers usually color manage document CMYK color values—if you print a CMYK file the document’s CMYK values get converted to new values via the printer output profile. There are Postscript RIPs that will let you output document CMYK values unchanged, but they are not common.
Also, Duotone mode assumes the output is to separated spot color inks for offset printing—Ink 1, Ink 2, etc. If you set your Info Panel’s eyedroppers to Actual Color and sample a Monotone, the readout will be ink 1. If you set the dropper to CMYK, the Info panel is showing the conversion from the Monotone spot color to CMYK—the converted CMYK values depend on your Color Settings Spot and CMYK Working Spaces: