I hope this feature will be implement into SS in the future
You could use the # of likes as part of the cache key. So rather than using LastEdited define your own key (http://doc.silverstripe.org/sapphire/en/reference/partial-caching#cache-key-calculated-in-controller) which uses page id + likes + last edited to generate a unique key.
Or you could load the like values using ajax which is fired off when the page loads. That request would return something like whether the user has liked it and the count of likes. Though looking at your example I can't see how you're passing back whether the user has voted or note.