additional Trunk link between 2 LTM load balancer in HA vlan

NO, mgmt interface not using for failover.

it is the best practice (to also setup network failover via management interface).

F5 recommends that you configure network failover to communicate over the management network, in addition to a TMM network. Configuring network failover in this way creates a reliable failover communication channel while the system initializes, and before the TMM network connectivity channels are fully established. To do this you must configure a failover unicast address and a failover multicast address for each device group member.

sol14135: Defining network resources for BIG-IP high-availability features (11.x)