Don't think CB is best for such a scenario unless a user profile is actually a Book profile. Instead I highly suggest using a direct extension in combination with CB to provide what you're wanting. There's quite a few great direction extensions below.

extensions.joomla.org/extensions/directory-a-documentation

With that said it's possible to do this if you really want it, but it's going to take CB Auto Actions and a silly combination of actions to accomplish it. You'd need to create a query action to check that the book fields were populated after registration and after profile edit. It'd then need to push that information to the Books database. You'd then need a field action that fires after that action (copy it, then change it to Field) that empties those fields. I've no idea how you'd search this database though, but you could display it I guess with CB Query Field.