CMYK, or process, builds will almost always have some color bleed through simply due to how process colors work.
You could disregard the visual color and use 100% Cyan, 100% Yellow, and 100% Magenta. After all three colors are three colors and on screen it makes little difference as to what those colors look like. When working this way, it's the ink on the press that determines color, not the image you see on screen. This would give you just three colors.
To do this you simply delete data from the channels you do not want that color on. For example, if setting up the Cyan, you would select the Yellow Channel and erase (paint white) anything which is not meant to be yellow. Then do the same for the Magenta Channel and Black Channel - erase anything which is not intended to print Magenta/Black. Repeat for the other colors.
This method works but the actual image on screen will not appear in the colors. You have to learn to trust the construction, not the colors on screen.
If you are dead set on seeing the image in the actual colors they will print in, you'll need to use Spot Color Channels. Spot color channels allow you to select a spot color (Pantone, Toyo, etc) and use a channel to display that color information.
Configuring spot channels can sometimes be a daunting task for someone unfamiliar with print production and color separations in general. The difficulty arises for many in the fact that you can't utilize layers within Photoshop when working this way. Everything has to be done on the Channels.
Here is a basic tutorial, from Layers Magazine, on configuring spot color channels.
Essentially you create a new channel, select what color that channel represents, then paint black on that channel. Black is where the ink will be printed. You then repeat the process for each separate color you want to use. Then save as .psd, .pdf, or .dcs to retain the color information.
Of note: If the entire purpose is to create artwork for laser cutting, Photoshop is really the wrong application to use. You should be using Illustrator or Corel Draw, or a vector application. For laser cutting you want strong, crisp, straight, lines. That's not what Photoshop excels at.