Fortunately after some research I found a way to retain this capability. In this article I'll describe how to upgrade the DSM software and retain the ability to use USB devices.
It is worth nothing that the support depends on the inclusion of the appropriate driver modules by Synology in the distribution. These are currently present, albeit unsupported according to Synology, so mileage may vary in the future. Should this change, then I will need to migrate my setup to a different device.
Let's see how we can do it for the time being.
Upgrade DSM to v7.0
Before you can start - you will need to download the update from Synology.
Once you downloaded the update, click on "Upgrade" and review the notices - you will see that they warn the discontinued support for USB devices. Fear not, and march forward!
And voilá ... the upgrade is now ongoing. Remember to avoid unplugging / switching off your Synology Disk Station during the upgrade to avoid any unnecessary pain.
When the process is complete, you will see the following message. I simply reloaded my browser to check the status.
In my case I ran into a couple of quirks:
My SSL certificate was removed. This was expected, since I was managing it out of band via a custom installation of Certbot. After some investigation, I figured out that my custom script was relying on commands that are no longer available in DSM 7.0. For example, the correct way to restart nginx (used by the DSM web interface - to pick up the updated certificates) is systemctl restart nginx and not /usr/syno/sbin/synoservicectl --restart nginx like before.
When refreshing the page (after accepting the security warning due to the expired certificate) I noticed that I kept being redirected to the login page even though the upgrade had not completed. I have simply refreshed the page until I saw the completion of the process, then logged in as usual with my credentials.
Add support for USB devices
In order to achieve this I strongly recommend to follow the approach described by LunaticMunch on Github - in short it relies on creating a startup service for loading the required drivers needed for recognizing correctly the Conbee II stick.
All that is needed is to enable ssh access to your Disk Station, then create the appropriate startup script as described in the repository I linked. Assuming you followed the same approach I described using the Deconz docker container to manage your Zigbee USB dongle, then after loading the drivers you will need to restart your Deconz container so that it can see the device correctly.
After following this approach and restarting the container, my prior configuration was picked up immediately and I was able to work with Home Assistant and my sensors as before the DSM upgrade. That was quick and easy!
I hope you will find this short tutorial useful - let me know if you ran into any problems!