Hi Yohann,
Thank you very much for the details. Unfortunately the functions.php file does not contain information about the search form on the site. It is very likely hard coded somewhere in the header template files – that is the reason the automatic replacement has no effect.
You might have to ask the theme author for instructions on how to replace it, or if you want, you can add temporary FTP details to your ticket and I will try to locate it and replace it for you.
Please note, that I will do my best to help you out, but I cannot guarantee a solution when making modifications to files.
Let me know!