Peter,
Your "Electric Lime Green" is out of gamut for CMYK. Bright RGB colors like these simply can't be printed using normal inks.
On the bottom is your original image, on the right is the image soft proofed to CMYK (US Web Coated (SWOP v2)
Notice how the green dies?
On the left is the image with Gamut Warning tuned on. Everything that is gray is out of gamut, even the red & blue of the flag.
The only thing that might be close is to print this on a press with a Fluorescent Green Pantone spot color ink. This is an ink made specifically for printing a bright green that cannot be printed any other way.
There are NO values you can use to print this using CMYK. Just ain't gonna happen!
Now your problem is trying to explain to the customer why what he sees on his screen can't be printed.
Good Luck with that!
