Return a file in my controller that is downloaded from an API - ajax

Question

I would like to return a file in my controller that is downloaded from an API (who needs a basic auth).
This API returns the file and the problem is I can't use ajax because of Access-Control-Allow-Origin, even when I add the basic auth in the header, it returns this error. And I don't have access to this API so I can't change it the config.
So I found another way! I call this API from my controller with the header and the basic auth. It's working! I can access this API and I don't have the error message: Access-Control-Allow-Origin.
But the problem is the API returns heavy files, and I have this exception OutOfMemoryException. It's normal.
So I would like to returns in my controller just a redirect link to this API with the header and basic auth to download the file. Do you think it's possible?
An example of what I want (But it's not working):
public function indexAction()
{
// API
$url = "http://api-who-callback-a-file.com/api/file";
//Create header
$curl = curl_init();
$url = sprintf("%s?%s", $url, http_build_query(false));
// Optional Authentication:
curl_setopt($curl, CURLOPT_HTTPAUTH, CURLAUTH_BASIC);
curl_setopt($curl, CURLOPT_USERPWD, "admin:admin");
curl_setopt($curl, CURLOPT_URL, $url);
curl_setopt($curl, CURLOPT_RETURNTRANSFER, 1);
// Get file (Problem here because the file is to heavy to be downloaded by the server)
$result = curl_exec($curl);
curl_close($curl);
// Return the file
return $this->file($result);
}
In this example, it's not working because the server download the file and it's too heavy. Me I just want to return the URL with the Header to the client, and he will download the file.
Thank you