If you get an error trying to copy the library via Finder to a larger SSD, that has nothing to do with FCP. That is a filesystem error that normally indicates the file is physically damaged. That could be the same underlying problem that kept FCP from backing up that library. If it can't read the library because files within the library are physically damaged, it can't back it up.
In such cases it's best to take a deep breath, slow down and don't make it worse. A good step would be search for all existing backups of that library, then copy those to a safe location. FCP automatically backs up libraries in a compressed format that doesn't include cache, analysis or media files. If you haven't used that library in 3-4 months, the may be several good backups that FCP automatically made back then.
The default library backup location is in ~/Movies/Final Cut Backups.localized. Each of those folders is a library name. Within each folder are often several backups of the library. Each of those backups is a copy of the library at a certain point in time. Each can be opened in FCP by File>Open Library>From Backups, or you can double click on the backup library in Finder.
Before doing that, as an extra safety step, I suggest copying all of the library backups to another "known good" drive. That preserves all the current backups in case problems get worse.
If you have any especially critical FCP or non-FCP content, you might consider manually copying that to a "known good" drive, just in case.
Once that's done you could run Disk Utility First Aid on the external drives where the library was. Look closely at the output. If there are any errors, I'd take a photo of that with your iPhone, just in case it's needed later.
First Aid does not do a read/write or sector scan of the media, it only checks for the filesystem structures. So it's possible First Aid could have no errors but the media itself could have bad blocks. Normally a failed read would be logged in the macOS System Log, but that is very verbose.
A simple test is running Blackmagic Disk Speed Test on each drive. One failure mode is the drive may show very slow write performance. Some SSD drives can enter a "Flash Translation Layer (FTL) failure state," in which writes become extremely slow. Both Samsung and SanDisk Extreme Portable and Extreme Pro drives have experienced this. I don't know about the PCIe versions.
Any media used with FCP should ideally be formatted APFS, both SSD and mechanical drives. The previous advice about APFS being only for SSD is out of date.
