If you were to create new categories, Beachfront would stay in the same place, but more categories would be added below it.
If you were to delete the old categories and create new ones, that would have the desired effect, but any posts that were associated with the old categories would end up having to be re-assigned to the appropriate categories (I think the posts automatically get re-assigned to the default category; which, in most cases, is “Uncategorized”).
You could conceivably change the category IDs, but a) the IDs are only changeable by manually changing items in the database and b) you run the potential risk of using an ID that’s already in use somewhere else. Therefore, I would definitely recommend against that.
Unfortunately, there is no way to tinker with a copy of the site (short of installing another WordPress and importing all of your settings, posts and files). However, you could use a plugin to backup your entire installation (find one that backs up your files and your database information) and then restore from that backup if something goes wrong.