"identify of the remote computer cannot be verified" certificate popup with internal FQDN of remote desktop gateway appears when connecting externally using RDWEB

I currently have Windows 2012 R2 Remote Desktop services configured with RDWEB. From the Internet, the user can openhttps://mypublicdomain.com/rdweb, login, and click the icon to initiate a Remote Desktop services session to one of my hosts. However, anytime the user clicks on this link, there is a pop-up stating"the identity of the remote computer cannot be verified". It lists the proper publicly trusted SSL certificate, *.mypublicdomain.com. However, the "requested remote computer" is remote_desktop_gateway.myinternaldomain.local. Since the certificate does not contain my internal domain name, this error keeps coming up. Is there way to change the "requested remote computer" to be an FQDN that matches the certificate so the gateway's FQDN does not appear?

Here's what my environment looks like:

1. Remote Desktop Services Gateway: RDSG1.myinternaldomain.local. This server also has remote desktop web access and I've published the public URL RDS.mypublicdomain.com with a trusted 3rd party SSL certificate (there's no trusted certificate for myinternaldomain.local)
2. Remote Desktop Services Connection Broker: remote_desktop_connection_broker.myinternaldomain.local
3. Remote Desktop Services Session hosts: RDSH1.myinternaldomain.local, RDSH2.myinternaldomain.local, RDSH3.myinternaldomain.local

Thanks!