Cannot add F5 net route

In general/recommended, you would use the tmsh commands on bigip to add the routes. and yes, tmsh sys management-route is another way of adding route entry specific to send traffic to a specific network via the mgmt interface.. the 'ip route' command will show all routes in the bigip, regardless if they are tmm or mgmt route..

here some reference that might be useful for this topic: K13284: Overview of management interface routing (11.x - 13.x)