How do I detach a copy of a project from its parent in Flow?

When you set up an action to open a Web Link (for example, a button element or some text in a text element that the learner clicks open the web link), the associated link is added to/included in the Resources list available under the navigation menu. That Resources list will also show any documents or other files that have been added as using the Attach Resources process. Because these are two different processes, the Attach Resources feature doesn't show/list web links, as you've noted. 

To remove the web link from the Resources list in the copied Project, you'll need to find the page/place where the weblink is set up as the target of an action and remove that action. 

One caveat - the Save As New process has two options, Duplicate or Share. If the copied course was made using the Share option, then its learning content will be shared with the original course so edits to the copied course will also affect the original course. Before you remove the action that opens the weblink, make sure to confirm in the Project Outline that the learning object it is in doesn't have a sideways-"Y" icon on the to the right of the LO name. That icon indicates that the Learning Object is shared.