Is there a way to re-sync my linkedin.com profile instead of doing just a 'one time import'?
It would be great if I could update my LinkedIn (primary) then hit a button to update my Stack Exchange profile.
No, there is no method of re-syncing a LinkedIn profile to your CV on Stack Overflow. We actually got rid of the "import from LinkedIn" feature entirely a couple months ago because LinkedIn changed the terms of service for their API, and we were no longer allowed to offer that integration.
Unless LinkedIn changes their terms of service again to be more like the old version, we won't be able to offer any import functionality, including an option to update / re-sync.