We just updated the name of our grid master HA members last week.  There was no impact to service as I recall.

As you said, if the certificate has only the old name, your browser will give warning messages that the name does not match.  HTTP Strict Transport Security will cause your browser to lock the site down and you will not be able to easily bypass this.  You could replace the SSL cert with a new SAN SSL Cert with the old and new names.  This way it will work with both names.  I also add the IP address to the SAN list so it can be used as a last resort.