Is the easiest solution to backup everything on a a huge backup drive then do the HDD upgrade to the NAS?
Depends on your needs.
A concrete example: a few months ago, one of my customers (an architectural design company of 30 employees) needed to upgrade the disks on their three disks RAID5 QNAP NAS. They were however in a work rush period, and couldn't afford the performance impact of a RAID 5 rebuild during the day. So, we had to do the swaps during weekends. Each disk swap required 12-14 hours to resync the mirror, so we ended up spreading the migration over the course of two or three weekends.
The final step required the whole raid to be taken offline for a number of hours - that's the step where the RAID capacity is actually increased.
Again, this was the procedure for QNAP. You have to check the exact procedure for your specific NAS model first (assuming they do allow inline capacity upgrades).
If your RAID has 10+ disks, then consider the amount of time it will take you do do the disk replacements, one disk at a time, waiting for a full RAID rebuild between each disk swaps... And that final step where the whole RAID will be offline for a number of hours. Can you afford that downtime? Is the NAS content mission-critical (i.e. used by a business), or just your personal collection of videos/photos/music?
With such a large array, I'd recommend having a complete up-to-date backup, scrapping and recreating a new RAID out of the new disks, then restoring the backup to the new RAID. If anything goes wrong with the restoration, then your plan B will be putting back all your original disks in the NAS, and fixing your backup / recovering any missing files, etc... So in a way, this gives you an extra security net in case something goes wrong. With an online upgrade, you will be dependent on your backup, AND the existing disks not failing mid-upgrade.