I am far from an expert on this subject, I am curious about the application - this sounds like a really stiff spring for it's size.
302 and 316 stainless are commonly used for springs, you should be able to find wire for sale somewhere. It won't rust, but it will be a real bear to wind.
A 6mm OD with 1mm wire means a 4mm ID, so you need to wind it on something like a 3mm mandrel. I doubt you will be able to wind 12mm + waste between centers with a 3mm mandrel on the lathe the way most folks do. You will need to rig up some kind of follow steady between the mandrel and the tool post to keep the mandrel from bending/breaking/popping out the center. Going to bet that the mandrel will still wind up and twist/break though.
For sure you will have to anneal the wire before winding. Stainless anneals around 2000 F, but both 302 and 316 work harden, so it will probably come out of the mandrel hard (they can't be heat treated).
Is there no way to change the design? You might just have to purchase a commercial spring
Maybe you could "thread" a rod of 316 with 90 degree "threads," then drill/ream out the core leaving you with a flat coil spring. You would have to figure out a way to round of the edges a bit to prevent breakage. Just tossing out an idea for an alternative, dunno if it would work.