Sorry for the split reply. Posting from my phone which is quite awkward.
I would let the SMART test complete and let us know the results. As others have said, you need to take a backup before doing anything. I would figure how to get SSH access to the NAS from a PC using Linux ... you could use an Ubuntu live CD. I could do this with my NAS using the root user combined with the admin users password e.g. from the terminal, SSH rooot@nas_host_name followed by the admin users password when prompted. You may need to enable SSH access access in the control panel somewhere. Once you have done this, you can copy the data to your third disk connected to your PC.
These NAS boxes usually use mdadm which is most common form of software raid in Linux. That means it is possible to pull a good disk from a mirrored raid array and having physically connected it to a PC, access the disk with a few commands.
I would suggest backing up, replacing the knackered disk with a new one and then schedule regular rsync backups to another NAS or PC running Linux/Rsync. Use your third disk (and broken disk if it comes good) to host the regular backups.
It may sound complicated, but it's fairly trivial for a careful and moderately experienced Linux user. There may be some outlay, but I am sure those near to you can help out. I actually have a soon to be spare Synology NAS box that would possibly be ideal as a target for your backups. If you are near to me, I have a spare PC I could loan to you with a Linux distro installed and even help you hands on. I am not far from Brummy land.
Let us know how you get on with the test.
What size disks are you using?