Could you give a bit more info .... or include some screenshots or the file?
I'm on SW 2014, so if you are using 2015 or 2016, screenshots, or parasolid or STP file could work.
It sounds like you want to do 2d cutting in a sheet material, like with a laser or router. So if you have multiple parts, I think you could easily get a 2d view of each part, so I'm probably missing something.
I just realized you might want all the pcs together in the assembly, like for making a jig or template. In that case, you could go to whatever projected view you want and save a dwg file. If you have any spline curves, that could be an issue for 2d CNC. Most of those machines use arcs.
CNC cut from illustrator? That's a brand new WTF on me!
Nearly all 2D CNC is done from DXF. Are you sure you need to go all the way to Illustrator?
Yeah, its a bit unlikely that Illustrator would be driving the CNC equipment, but I do know the sign industry uses applications that are Adobe Illustrator compatible, and can generate files for CNC equipment (FlexiSign for example). So its quite possible someone would work in Adobe Illustrator and give the Illustrator file to their cutting service for router, laser or waterjet cut parts.
Then we are back to the spline to arc question, which might be handled without issue by the cutting service. However, I still hear of compatibility problems with splines, which suggests that some services can't effectively convert them, or don't want to take any liability for the conversion.