My GP told me last time that Riley had a gastric thing to keep food up to him as it puses the bug out of his system faster. She also said no dairy but bananas were good and so was bread, plain biscuits. She told me to keep him off high fibre things too. Your milk won't make him sicker, it will give him antibodies that you are building.
I would get the script done if he has green snot as this shows infection.
Can you get him some baby dimetapp or demazin or Fess saline nasal spray? I find that helps my kids.
Good luck, its very tiring having a sick little one especially if you are sick too.