> Different kinds of claims

How so?

If I claim that one should prefer the claim "moral knowledge doesn't exist" over its contrary, then I am making a moral claim. That would make it self-refuting.

There is no fact-value dichotomy.

And one more thing...

> the lack of falsifiability

Is falsifiability falsifiable? If all credible claims must be falsifiable, then where does that leave us with the criterion of falsifiability (which is problematic even part from this particular case, as anyone who has done any serious reading in the philosophy of science knows).