GTM w/o BIND and NS Records

As you say, the client is behaving oddly. It's unclear when it "thinks" it has finished following the delegation chain if it is proactively requesting NS records. But, assuming it wants the delegatee NS records to match the delegator NS records, then you can use ZoneRunner to insert or modify the NS set on the BIG-IP for the zone in question to match those provided by the parent.