Hey faddoo
It seems that you actually already have your solution. On your first followup you provided the code for a CURL function that retrieves the JSOn feed with those headers. You could simply make a script out of that, which will output the contents of the retrieved json feed.
The simply use the URL to that script as the data source for the import. That way the import is already getting the authenticated feed, while the authentication is handled by your custom script.