Without asking lots more questions, as I don't have a QNAP to compare...
Don't use the clone library you created. Use the same library whether you are local or remote.
Set up a VPN to your Public IP Address, which means to your router, if it supports a VPN connection to it in the firmware. You will need a VPN Client application to run on your laptop. Your router supplier may provide one for free, or you will have to find one that works. Or use the VPN connection available in Windows. (See the web page I linked to on the QNAP site below for information on how to do that.)
Once connected via the VPN, you should be able to see all your LAN resources using the laptop when you are remote. Therefore, MC should run normally. You will need to check that the NAS address or drive designation is the same when connected locally and via the VPN.
If your router doesn't support a VPN connection, but the QNAP NAS does, you can set up a VPN to the NAS.
You will need to Port Forward the port that the VPN uses to connect from the outside world. As you can already see the NAS files using a web browser, the NAS probably already has port forwarding set up in your router to direct traffic sent to the external IP Address to the NAS. The QNAP Cloud function may use the same, or a different port. In fact they are probably both using the Web Server on the QNAP. However, if neither of those use a VPN connection, then you will need to set up a VPN Server on the NAS, and connect to that. Again, you may need to do Port Forwarding in your router for the VPN service. Here is a description of how to set up a VPN to a QNAP NAS.
https://www.qnap.com/i/au/tutorial/con_show.php?op=showone&cid=3Once you are connected via the VPN to the QNAP from a remote location, you will at least be able to see the NAS files, if not all resources on your LAN. Again, check that the NAS address or drive designation is the same when connected locally and via the VPN.
MC should then work normally whether local or remote.
PS: You could have set up a MC Client library on your laptop and access to MC remotely, if you had a MC Server running at home. But you don't. If you did have a MC Server running locally, your laptop would always be connecting as a Client. That seems unnecessary in your situation.
PPS: While theoretically you could do any work on MC while remote, including tagging, Cover Art, moving, renaming, and copying files, building and modifying views etc. (i.e. Lots of things you can't do when using a MC Client), you probably shouldn't do all that stuff over a VPN, as you are relying on an internet connection. But it would work, unless you lost connection. Which could be bad, with partial updates to tags and files, for example.