You need to put the SmartThings hub into inclusion mode and then with the Z-Stick and Zensys Tools put the Z-Stick into Learning mode. It is the little lighting bolt icon. After the Z-Stick is a secondary controller, you will have to request updates from the SmartThings hub whenever you add a device to your network from the hub (as @JDRoberts mentioned) . Looks like this: 
At this point, updating the Multi Sensor 6 is a little different depending on how it was joined to the network. If it was joined while USB powered, then you just find the node in the upgrade app provided by Aeon and click the update button. If it was joined while battery powered, you need to temporarily set the sensor to send its wake report to the Z-Stick. The code for the SmartThings device handler would be (replace nodeid with the id of your Z-Stick - in hex):
zwave.wakeUpV2.wakeUpIntervalSet(seconds: 300, nodeid:55),
You may also need to associate the sensor to the Z-Stick (I can’t remember at the moment):
zwave.associationV2.associationSet(groupingIdentifier:1, nodeId:55)
Then, when you hit the update button in the Aeon firmware update tool, the update tool will tell you that it is waiting for the sensor to wake up. Press the button on the back of the sensor and the firmware update should proceed.
Which device handler are you using for the Multi Sensor 6. If you are using mine I can guide you more regarding where to put those lines of code.