The printer is probably more familiar than you with how to do this, so you may want to just send an Illustrator file.
If your Illustrator file has been pared down to only the four spot colors, I think it will print separations properly. Illustrator's choose-by-color capability should allow you to root out anything that's still an RGB or CMYK color and make it one of your four spots. The main concern for you is which colors should be set to overprint. Without thinking intensely about it, I would say all of them should be. Silkscreen inks are pretty opaque.
If spot colors don't give you four, and exactly four, separations with no screens other than the ones you deliberately put in for the open water and woodland fill, you may have to resort to the way we did it in the 90s: having the CMYK seps pretend to be spot color seps. Make contours and contour labels 100M, and then tell the printer the M separation prints in brown. Make the green stuff 100Y and [some percentage] Y and tell the printer the Y is his green sep. Make the water 100C, etc.