I've done hundreds of night shifts in my time and you have nothing but my sympathy - they are completely unnatural, i.e. having to stay awake when it's pitch black when most other people are doing what their bodies are telling them - sleeping!
Unfortunately, if your contract stipulates that you have to do them, it sounds very much as if they've got you by the short and curlies. You could always have a word with your boss/supervisor or whoever and find out if there's any way you could stick to the days instead? Worth a try.
The only thing about a doctor's note - but I do stand corrected here - is that they only last so long, and then you have to get it renewed, but if your doctor thinks you're just swinging the lead, you'll be signed off and back to square one.