GTM persistence in proxy environment

This isn't an unusual problem. At its core, the issue is that there's no information in the DNS request to allow the GTM to relate two queries from the same client that arrived via two different local DNS servers, although there are extensions that propose to solve this problem that aren't yet well supported in clients (EDNS0 client_subnet).

With regard to active-standby, if the request arrives at the GTM process, then it will handle it. In order to arrive there, it needs to be received by a listener, so if the listener is on a floating IP, then only the active BigIP will ever receive the DNS request.

Is there a reason you can't point your clients directly at the GTM listener IP addresses, and enable recursion on the GTM ? This would allow the GTM to see the original client IP address, handle the request if appropriate, otherwise recurse it.