Just my opinion, but it sounds like you need to build a plugin that connects to your site and also look at the embedding process.
Your plugin would need to restrict the like button based on the result it gets from your site and run an ajax call back to your site to keep it updated. I would probably run a couple daily sidekiq jobs to make sure everything is in line.
But to answer your question, yes. I would say it’s possible. But is it worth it?