First idea - can't you just send an ink cartridge to the lab? That would be easier than what follows.
Most of the printers I know of don't have a way to directly specify that you want to use light Cyan vs. regular Cyan. These printers generally take CMYK input (or even RGB) and internally they do separation/screening to CcMmYK.
What you should do is to create a set of swatches of varying tint values (ex: 50c 45c 40c 35c 30c etc.) and print these. Examine them with a high powered loupe to see if you can see the difference between Cyan dots and Light Cyan dots. If you can't see it, try creating patterns like concentric squares with the dark tint as the outside square and the light square as the inside one. Once you know what tint percent produces Light Cyan and Light Magenta, you're set.
Note 1: Many printers have unique halftone screens for each media type, so make sure you do this experiment on the final media.
Note 2: You may also have to figure out how to print with color management turned completely off, not only in Illustrator, but also in the printer. With color management on, there is a good chance that a Cyan swatch in Illustrator will not print as only Cyan. If there is no way to turn it off then you will need to experiment with different RGB values in Illustrator to see if you can find one that prints as pure Cyan.
These printers generally take CMYK input...
I must respectfully disagree. With the exception of professional-grade models with PostScript RIPs, inkjet printers, on the whole, are RGB devices. You can prove this to yourself by examining an ICC profile for a typical desktop inkjet with a text editor: early in the file you'll invariably find the identifier "prtrRGB." This is counterintuitive because these things use CMYK-based inks, but that's the way it is. Whether it's handled in the driver or firmware (or both), there is, effectively, a space conversion from the input RGB data to that required for the subtractive-color ink output.
This is all the more reason individual ink tank output control remains beyond the reach of end users. I agree, therefore, that sending actual ink samples for testing is the best method.
This is also why one should work in RGB document mode in Illustrator if the intended output device is a typical inkjet printer (rather than, say, a printing press or a PostScript proofer).
No, they are not RGB devices. Think about that for a second and you will realize that is a error in perception. It is a CMYK device that is driven by RGB input but the device's drivers and firmware process the rgb data for printing based on data receive through the drivers about that conversion. It also depends on the medias profile as well.
They are getting very good at this so it is good to understand that the info sent to the drivers are different fro the info the drivers and firmware send to the printer.
I doubt you can manually create a dedicated swatch that will print 1:1, but your printer driver should offer a test print function that certainly should be usable for this. If you haven't already, you should also start by downloading specific color profiles for the printer, assuming they exist. This should to some degree allow you to enforce certain colors. The long way round might also be to disable individual cartridges in the printer driver, which the more expensive models usually allow to keep on printing even if you run out on one ink. The rest may take some experimentation until you get a suitable print...