How do you make a cold iron rapier?
First off, iron and "cold-iron" are the same. There is no difference. Cold Iron is simply an iron implement designed to cut. "Cold Steel" implies the same -- and means the same - except in relation to steel.
Iron is Fe without the carbon. Steel is Fe with .3 to 1.7% carbon (and is heated differently depending on the carbon component of the steel).
Now, how do you make a rapier ouf of iron that has no carbon in it and is not annealed? You don't.
There is no such thing -- and no amount of "fantasy" should allow it, either. You want a rapier? It must be steel, not iron.
The rapier is the high-point of sword/knife mettalurgy in Renaissance Europe. It is the functional technological equivalent of the Japanese katana. While there were differences in each in terms of how they were made, the flexibility of both the katana and the rapier was permitted by annealing the steal differently at different locations on the blade. You can't do this with iron and achieve a variable temper on the blade.
What makes a rapier a rapier is not its edge, it is its harder back/core and softer edge, giving the rapier an ability to bend as spring steel.
It's too thin to be made of iron and iron is too inflexible to be fashioned into a rapier. It will snap. Every time.
Sorry. You can't get there from here.
Sometimes a GM needs to say "no." This is one of those times.