Well, this is both encouraging and confusing! While booted up via USB, can you successfully run First Aid on the drive, or does it emit the same error message?
I share your concern that a CCC backup would copy whatever the problem is. But, if you can run CCC to back up the drive while it's attached via USB, then you should be able to re-install the OS. Migration Assistant should run during setup, and I believe it'll be able to read from the CCC backup to restore (only) your files and settings.
Only you can decide, but after a successful CCC backup, the first thing I would do would be to attach the bad drive to your wife's computer and use Disk Utility to erase it. Erase the actual device, if you can, not just the existing HFS partition. After successfully erasing, run First Aid as before, just to make sure the drive is working properly. If it looks ok, put the drive back into your laptop, then boot from your Sierra installer USB and go through the installation. The drive will be already erased, so perhaps you can skip that. When it finally runs Migration Assistant during the setup, attach your CCC drive and let MA recover your files and settings.
All of the above with fingers firmly crossed, of course. It's still not clear whether the problem is "just" with the drive, or if the disk controller is involved. But we live in hope, and the drive is the more likely culprit. Sorry for so much text, I hope it's clear.