Since Paracord (if genuine) is Nylon, then it's generally safe in this situation. Nylon is, after all, used in food packaging.

Nylon is mold resistant and does not rot.

If you do go this route, just gut the paracord and use one of the inner nylon core threads. They're pretty thin and would work here.

All of that said, just replace the chain? Chain is cheaper than paracord anyway.

https://en.wikipedia.org/wiki/Nylon#Characteristics

If it's actual paracord, then it's made of Nylon. Nylon neither rots, nor supports mold growth. It also dries much quicker than other materials.

It is highly resistant to molding, but it's not impossible, and it is used in food packaging.

https://en.wikipedia.org/wiki/Nylon#Characteristics