I therefore hold that it is inconsistent to be relativistic about morality but not about empiricism
Well, then it's also inconsistent to be relativistic about gastronomy. And wine. And fashion. And books. And prettiness.
If you say A, you've got to go through the whole alphabet :-)
On the other side, of course, is what is basically Samuel Johnson's refutation. You want to deny empirical reality and science, maybe I'll even come to your funeral. You want to deny some particular ethics, well, what will happen?
Short answer, people will kill you. The long answer is about 2.5 sheets by now. Maybe I'll post it :). Hopefully that won't go as disastrously, with people getting pissed off, as this one.
A couple of years back I stumbled across this diagram on reddit. Since then I've kept a list of all illogical arguments, quackery, irrational ideas and similar nonsense, to pass the time. The count as of today stands at 1229 irrational ideas (a few could be debated perhaps). Hopefully you'll have a laugh or two! Any additions let me know!
Link to the list:
https://docs.google.com/spreadsheets/d/1T3HQ6TnCg6Q44WzpJPFxLvfqbfvKmT5RqvpGI3ji1y8/edit?usp=sharing
And the original diagram: