Your 6 colour printer most likely separates from RGB to its own colours via ICC profile. If you send it CMYK data, it'll translate that first to RGB, and then generate the separations with its own 6 colour system.
However, not all RGB colour spaces are the same. The ICC "PCS" or profile connection space tends to use a different set of values for its component colours than those used by most RGB profiles, which are based on the colours that screens render easily. The sRGB colour space is horrible, really washed out, being a worst-case-scenario estimate of what a screen can do.
Best file format to save in is this case is an RGB illustrator PDF with an Adobe RGB profile.
You could get into calibrating your screen, and embedding the profiles for your printer into your documents, but I've found that by and large Adobe RGB gets better results. Every time you narrow the colour space, you lose quality.