This isn't a fundamental problem with the idea here, but one thing I don't like is that it makes hashcash.io a point of failure in your form submission. If the site goes down, either you can fail open (spammers get to comment) or closed (nobody gets to comment), but either way you fail. It's also going to add additional latency to your form submission.

I don't see any reason why this needs a third party, although I do grant that makes it easy to use.