added to the bridge API in hivemind, and then the browser extension would make a bridge API call to a steem node?

Right. Like the new get_bookmarked_posts here

Would a soft fork be needed?

No, only an update of hivemind nodes. We don't even have to wait for Steemit to integrate this into their nodes, as we can also use the extension to query our own nodes (e.g. the one on which my database version 21 with the bookmarks is already running).