You can stop Hiver from syncing existing emails in a shared label by switching off the option to sync old email under advanced options for the shared label. Just edit the shared label, go to advanced options, and switch it off.

While setting up the sharing for a label with a large number of existing emails that you don’t want to share, just switch this option off when you’re setting up the shared label. Once that is done, only the emails that are added to the label after the shared label is set up will be shared.