LTM as balancer of pool of DNS servers

There is really only one way to do this from a network perspective.  If you put an interface/IP on the F5 in the 2.2.2.0/24 network.  Then you would have to make the default gateway of the DNS servers be the IP address created in 2.2.2.0/24. For further HA you would need 3 IP addresses in 2.2.2.0/24, one for each F5 LTM and then a floating address.  The default gateway would be the floating address.  This will force all the traffic going to the DNS boxes to come back through the LTM to keep the TCP Handshakes functional.  

To further complicate I have done scenarios where the DNS box has multiple routes on it.  The default route goes to the BIG-IP LTM but then other routes for internal clients to a different router.  But to accomplish exactly what you have asked I would use above method and make the LTM the default gateway of the DNS boxes.

What about using EDNS0 Client Subnet?

Yep, seems to me that EDNS0 is the way to go as described RFC7871. Unfortunately, my HA boxes doesn't have GTM.