BIG-IP DNS with LTM integration

You can certainly use private IP addressing on the LTM, I for one would prefer this design. iQuery messages are exchanged using self IP addresses of each F5 device, these are used to provide status information, metrics etc on the downstream LTMs.

The only question you need to ask, regarding public IP addressing is if your DNS/GTM listeners are public IP addresses or private. Using public is simpler, however it it totally possible to use private addressing as there exists the capability to define the NAT IP when configuring your system.