There are two possible ways of doing it in Photoshop:

• using CMYK mode of printer
• using RGB mode of printer

Either can be unsupported depending on the model.

Using CMYK:

• convert to BW using any method you'd like (desaturation is the simpliest);
• switch the mode from RGB to CMYK (Image->Mode), use any suggested profile;
• use channel mixer with following values respectably: {0, 0, 0, 100; 0, 0, 0, 0; 0, 0, 0, 0} (the only positive value is 100 for "Black" in "Cyan" output channel)
• disable colour management and all adjustments in your printer settings completely (this may be tricky, one should pick the settings which are used for profiling)
• pick "Printer manages colours" in Photoshop
• print

The choise of CMYK profile will affect tonality. Pick different CMYK profile or adjust it with curves for liking.

Using RGB:

• invert
• use channel mixer with following values: {33, 33, 33; 0, 0, 0; 0, 0, 0} (red output channel has equal amount of every input channel)
• invert
• disable colour management and all adjustments in your printer settings completely (this may be tricky, one should pick the settings which are used for profiling)
• pick "Printer manages colours" in Photoshop
• print

The result will depend on the gamma of RGB colour space which is in use. Again, pick one which is better.

It is likely that you will get pure inks in either way but it is not guaranteed. These actions can be scripted.