I have updated a bunch of tags on all my movies and I would like to copy all the sidecar files to my backup drive. I just want to overwrite the sidecar files on my backup drive, not recopy all the movie files.
If you do the backup via MC itself (via Rename, Move, and Copy) and have it import the files into a MC Library, then MC itself can update the Sidecar files easily via the normal mechanism. But, I'm a bit confused over what exactly you're looking to accomplish, and I think that MrC's suggestion is actually a better fit for your needs.
If the goal is to have a backup copy of media files, along with their associated sidecar files, kept up to date as you make tagging changes (especially things like [Last Played] which might change at any time), then your best bet is to implement a real Sync solution for backup. If you aren't as wizardly as MrC with the command line, there are a ton of great tools for things like this. Here are a few:
GoodSync: Cross platform, powerful, and flexible, but the UI is a little clunky.
SyncBack: Very simple to use and powerful, but Windows only.
SyncToy: Free and somewhat limited, from Microsoft. But free, and might solve your issue.
Or, perhaps just good imaging/backup software. In this space, the answer is easy, as
There Is Only One that I'd even consider recommending anymore:
Macrium ReflectIf you need to make images and backups, automatically, and be able to schedule them recover using a Preboot environment (recovery disk) type of setup? Just Get Macrium. Like Acronis before they made it crappy, but honestly better than Acronis (or Norton Ghost) ever was in the real world. A fantastic product.
NOTE: I use all of the products listed above for my personal needs, except for SyncToy (which I used to use, but outgrew quickly). I use, and LOVE, many things about Goodsync. I just wish the UI wasn't so clunky. It works great if you need to create something like "your own Dropbox" where files automatically sync back and forth between machines, including different platforms, as the files change (not on a schedule). It works pretty darn well, and has a lot of power. However, for ease-of-use and reliability, I love SyncBack for my automated (scheduled) backups.
For System Imaging (compressed, restore a partition style) backups, then I use Macrium Reflect, and it simply cannot be beat. And, it has lots of File Sync capabilities too, so it might serve all of your needs.