I'd forgotten about this one - needs updating to use bootstrap one day, but still seems to work just about.
These are still jQuery-UI dialogs so you need to refer to the API here: https://api.jqueryui.com/dialog/#method-open
So if the dialog is the first one on the page you could use this to open it:
$("#wpdialog_1").dialog( "open" );
If you get this working and can add anything more to the doc page that would be great!