FCP error: The Storage location for one or more libraries is no longer available. The libraries will be closed so that no work is lost

Please go and check to ensure that Final Cut Pro has Full Disk Access in System Settings->Privacy & Security->Full Disk Access

That's a backup library. Are you really meaning to work in that, not the original library? The storage location for that is normally the Final Cut Backups folder in the user Movies folder.

What drive is disconnecting? How is it formatted and connected?

Please post an EtreCheck report https://etrecheck.com giving the application Full Disk Access in Privacy & Security System Settings. Copy the report and use the Add Text button to paste the result into reply box. 

You have chrome and its evil keystone agents and daemons. They are poisonous to video applications.

See https://chromeisbad.com for an explanation and instructions for removal.

After that you’ll need to delete FCP and download a new copy from the App Store.

In addition to the always excellent advice from Tom and Luis, after removing the nasty keystone stuff and reinstalling FCP, I would recommend not putting your backups on the same drive as the library. If the drive fails (and maybe it is beginning to, that's why it keeps dismounting), you'll lose your library AND the backups. Etrecheck shows you don't have TimeMachine setup, so unless you've configured another backup method, you run the risk of losing all your work.

Is the drive being connected through any kind of hub? The same cable is used to connect the drive for both machines? Can you try another cable?

Also, please post another Etrecheck to make sure you've deleted the keystone stuff.

I've had a spinning drive with a bad port before that would disconnect itself when it was furiously writing or reading. This reminds me of that. I ended up putting it atop some packing foam and gently taping the cable down, while I made a copy of it to stabilize it somewhat.

In addition to what Tom said: make sure that FCP has Full Disk Access in System Settings->Privacy & Security.

Has the backup library been moved to the LaCie? What is "Sorted Clips?" A project?

What exactly happened to the drive? How is it connected?

Please post the EtreCheck report.