Ok, I found a way, I think the issue was that it's a remote server. I found I can get around this using by claiming the remote file is JSON. It still returns an error if I check for it, but it works

Code:
callURL = 'http://remotedomain/setcookie.php?ProfileID=' + ProfileID;
$.ajax({
  url: callURL,
  dataType: 'jsonp',
  success:function(response){
    alert('Yay!');
  },
  error:function(XMLHttpRequest, textStatus, errorThrown){
  alert('Nay!');
}
});