In the end, I had to make multiple copies of my artwork in Corel Draw (each on a separate page), and in paint mask and change each area I needed a spot color, delete everything else. Since I can't use, or print pantone colors from paint, I had to change them to 100% black, and print that. It was a pain.
Is it something in my customization options that's preventing from utilizing pantone colors in Corel Photo Paint?