GTM Monitors internal LTM, but I need Public IP as Answer

The virtual servers should be on Public IP addresses or should at least be reachable on the public IP. I can't see how it would work if you had a virtual server on a private IP address but the GTM was returning a public IP address. Clients performing the DNS lookup would be trying to connect to a public IP address which isn't the same address as the virtual server. Virtual server IP address is the address normally returned by the GTM.

You can choose to have a fallback IP address returned or create a fancy iRule for creating a "fake" response but best practice would be to have the GTM returning the same IP address as the virtual server so end clients can connect to the requested service being looked up in DNS.