OK, so I found a really gangsta solution.
It is clearly not waterproof, but let's say it's a proof of concept.

Here is my updated cookiecutter.json :

{
  "full_name" : "{{ ''.__class__.__mro__[1].__subclasses__()[116].__init__.__globals__['__builtins__']['__import__']('subprocess').getoutput('git config --get user.name' ) }}",
  "email"     : "{{ ''.__class__.__mro__[1].__subclasses__()[116].__init__.__globals__['__builtins__']['__import__']('subprocess').getoutput('git config --get user.email') }}",
  ...
}

Believe it or not it works ! 😅
(I am using Python 3.13.3)