It might help if you use Ink Manager to Convert Spots to Process...
But even that is going to be problematic. Many spots have no good process equivalent, and your printer sounds like an RGB device which means it will accept CMYK input, but will convert to RGB, then back to CMYK internally when you print, so you have more potential color shifts.
I've used Ink Manager to convert the spots to process, and I've gotten the same result more or less.
It feels like I'm missing a crucial step in setting it up color wise??? IDK...
In any event, thanks for your input, Peter.
Do you have an output profile for the printer? If not, you might want to also try going back to using the spots unchanged, but set the print dialog to let the printer manage colors.
You should also try printing from an exported PDF.