Let me try to answer your question, since I’m using HyperCache myself a lot.
As my understanding, HyperCache some how should find that your page, which was rated or liked, became old and has no need in cache system such HyperCache.
So, you should see, which cookie is set when user “like” or “rate”, then turn on option “Bypass the cache for readers with cookies” with it.
Or, you may try to turn on ajax rating update in your plugin. Since it doesn’t use caching, it won’t be affected by HyperCache.
IMHO B-)