Setting up RDS using an external FQDN

We currently have a Windows Server 2012 Standard setup as a RDSH which works just fine for internal users. We point them tohttps://srv-rds.mydomain.local/rdweb and they can logon to the Work Resources Portal and are presented with the RemoteApps, which work just fine. We are using a certificate set to "srv-rds.mydomain.local".

We would now like to have RDS for use externally. We have setup a FQDN (srv-rds.mydomain.com) and setup our firewall with a reverse-proxy which forwards all requests to the RDSH. We are able to logon to the Work Resources Portal throughhttps://srv-rds.mydomain.com/rdweb and the RemoteApps are presented to us. However, none of the RemoteApps work. I have noticed that when connecting to the RemoteApp, the "remotecomputer" name is "srv-rds.mydomain.local" and the "Gatewayserver" is "srv-rds.mydomain.com".

Any ideas what we can do here, without affecting the current internal terminal server users?

Thanks for your help.