The load balancers aren't session directory aware in the sense that you are describing. BigIP can persist on the rdp username, assuming it is sent in the first request, or it can persist using the token, but it will send the traffic to the server (if available) either way.
You could disable the terminal service on your server while performing maintenance, this will cause the monitor on the BigIP to timeout, and it will not send any more connections.
The other options you have would need change on the BigIP, and since you don't have that access...
In our environment we give operator role access to the platform teams so they can enable/disable the terminal servers on their own for maintenance windows.