Perhaps you can create a custom URL parameter to use in your iframe src and then add javascript to DocView.aspx such that if it sees that parameter, it will perform the full screen button click.

In your iframe src, use something like

https://ourweblinksite/1/doc/123456/Page1.aspx?fs=1

In DocView.aspx, add the following before the closing body tag

<script type="text/javascript">
  var urlstring = window.location.href;
  var fullscreen = "fs=1";
  if (urlstring.indexOf(fullscreen) > -1) {
    $(document).ready(function() {document.getElementById("FullScreenButton").click()});
  }
</script>

See if something like that may work for you.