Copy product lines, into their own team or into another one.

Every requested copy is created: a name_suffix is numbered until it is free (next_available_name), so "Duplicate" twice gives two copies rather than silently doing nothing the second time. The created names come back because numbering can have moved them off the predictable one.

A caller copying a single product line may name it outright instead, and then a collision is theirs to resolve rather than the server's to work around: they can see and edit the name they chose.