Here was the solution:
Remove the tape server from Veeam entirely, then re-add it afterward. All tapes will appear in an offline state until the reintroduction and scan of the server, and I expect one of the two results:
- The library will show up as a new one, and the current one remains as an offline library
- The current library is brought back online with no duplicate
To do this, just go to Tape Infrastructure in Veeam B&R, expand the Servers node in the tree at left, right-click the tape server (which I understand to be the Veeam server itself) and select "Remove". Then, add the server again as if new, and run an inventory (not catalog) of the library.