XML Source Timeout Script Task

I'm new to SSIS and C# so I'm still getting to grips with all of this.

We pull XML data over a HTTPS link however when the data becomes too big the connection times out. 

After researching online I came across a suggestion using a Script task to retrieve the XML data and save it to a local file or in a package variable, which I can then read into the XML Source. I have tried to implement it by adding a Script task before the Data Flow task and pass the XML data to a package variable.

From a website online, I added this C# script in the Script task to pass the XML data to a package variable:

#region Namespaces
using System;
using System.Net; //Added
using Microsoft.SqlServer.Dts.Runtime; //Added
#endregion
namespace ST_3831315ea9bb45e9b1c2f2e9addad540
{
 [Microsoft.SqlServer.Dts.Tasks.ScriptTask.SSISScriptTaskEntryPointAttribute]
 public partial class ScriptMain : Microsoft.SqlServer.Dts.Tasks.ScriptTask.VSTARTScriptObjectModelBase
 {
        public void Main()
        {
            // Declare variable that will hold the xml document received by the API
            string xmlDoc = String.Empty;
            try
            {
                // Get the URI from the variable
                string url = Dts.Variables["User::URI"].Value.ToString();
                //Create a Web Client
                using (WebClient client = new WebClient())
                {
                    //Download the xml document as a string
                    xmlDoc = client.DownloadString(url);
                }
                // Set the value of the xmlDocument to the string that was just downloaded
                Dts.Variables["User::xmlDoc"].Value = xmlDoc;
                Dts.TaskResult = (int)ScriptResults.Success;
            }
            catch
            {
                 Dts.TaskResult = (int)ScriptResults.Failure;
            }
        }
        #region ScriptResults declaration
        /// <summary style="box-sizing:inherit;">
        /// This enum provides a convenient shorthand within the scope of this class for setting the
        /// result of the script.
        /// 
        /// This code was generated automatically.
        /// </summary>
        enum ScriptResults
        {
            Success = Microsoft.SqlServer.Dts.Runtime.DTSExecResult.Success,
            Failure = Microsoft.SqlServer.Dts.Runtime.DTSExecResult.Failure
        };
        #endregion
 }
}

However the Script task fails because of the same issue of it timing out due to data size.

How can I implement a solution so the Script task does not time out? Or is there another way around this?

Thanks in advance.