DNS/GTM for external use

You have answered your own question, NAT plays a major role here.

You cannnot reach the private range from the Internet. You'll need to NAT your private IP to your public IP.

Build your DNS setup, make sure its listeners are configured as NS properly for DNS resolutions. Refer the DNS build articles. Add the servers (LTM) with iquery. Because if iquery is setup properly & working, then DNS resolution for external to internal mapping should also work.

You'll have to put the actual public facing address as destination address & in the NAT field define your internal address. This requires co-ordination with your internal team, to know the routes, firewall, range etc.