The 'Copy the entire folder' to the new Computer is the only proper way to get success.
Never` 'Add' (or Remove/tamper) with the valid contents of a Library folder from the OS. AND Thrice never, overlay the contents (empty and replace is OK if complete Library contents)
Note You CAN change the main
Library Folder name as part the copy, You will need to use the Switch/Create menu to learn the new name (location)
The Library name is the Parent Folder Name
Old PC: Calibre Library
The New PC already has one (having 2 would be confusing, but you can

, as long as they are on different paths)
New PC:
(the old one pasted)Calibre Library=>Old PC Library <(the name is your choice)