Thank you all for your help. All I needed to do was set the StartInfo.WorkingDirectory to somewhere that I was able to write.
using (var proc = new Process())
{
proc.StartInfo.FileName = Server.MapPath("~/Testing/Demo/MyEXE.exe");
proc.StartInfo.Arguments = String.Format("\"{0}\"", commandFile);
proc.StartInfo.WorkingDirectory = savePath;
proc.Start();
proc.WaitForExit();
}
This causes the temp files to be written to a non-system folder and thus does not need any elevated permissions for the application pool.