Hello Thibaud,
Thanks for the updates. Yes, it is possible to design a normal page using Cornerstone and then you can get the page html source code and use it in your PHP code to create the archive template for your custom post type. Please keep in mind that this particular customization is outside the scope of our support as this is not related to an issue with the theme and instead has to do with your customization of it. As such, you will need to investigate this particular issue on your own or seek help from a developer should you not feel comfortable making these changes yourself.
If you have any further questions about the theme, we are more than happy to provide you with assistance on these inquiries.
Regards.