I have always hounded the vendors that Licensing is not something I need. I don't need to prove to myself I bought the license. They need me to prove it to them. So, the effort and cost should be on them. That said, this new requirement of licensing for NetScalers is causing more troubles than it should.
I can understand the position you are in. Any NetScaler Console VM or Agent VM will proxy the licensing to the designated License Server Agent, (including the built-in agent) and another Agent can be promoted to the LSA at anytime the first is missing for NetScaler Console Service. The problem is you would need to change the license server IP manually. As you stated, the device is already rebooted and not licensed, so pointing it to a new agent/LSA and rebooting should not cause an issue. If you already have the Agents or Local Consoles setup in HA, you should be able to point any NetScaler that is rebooted to the running Agent/Console and reboot, still negating the need for "HA" licensing.
The biggest pain in my view on any of this right now is the very possible loss of configuration/certificates during the brief "unlicensed" period.
All of this has been brought to the attention of the product team and I am hopeful they are working on better processes to make licensing easier and less obtrusive.