My thinking is something like this:
It is difficult to imagine a scenario where logic would dictate that you eat dog meat. Unless you were to construct an extreme scenario, I don’t know, a man puts a gun to your head and tells you to eat some dog meat or be shot, where it is unlikely that you wouldn’t eat the dog meat.

But when we get rid of such silly examples, what you’ve got comes down to taste.
If I happen to like the taste of pork then there is no rational reason for me not to eat it (lets assume the vegetarians are wrong for purposes of simplicity)
If however, I find pork to taste disgusting, then (barring extreme examples) I have no rational reason to eat it either.

That’s why I think it’s important that when you consider something like moral relativism, you emphasise the ‘moral’ part of that phrase. We are trying to consider whether morality is itself relative, having accepted perhaps that some things are and other things are not relative. We can perhaps separate your gut reaction to eating dog meat, and your gut reaction to something like female suffrage by saying that one comes down to a subjective opinion (which cannot be ‘right’ or ‘wrong’), while the latter can be tied to some sort of objective reality where we argue that it is morally/rationally correct to allow women in a society to vote.

Personally I believe I see the dangers of moral relativism, which is why Kant’s approach appealed to me last year. Unless I’m mistaken, his view suggests that the rational thing to do (which is objective) is essentially always the same thing as the morally correct thing to do. I’m pretty sure Utilitarianism has a similar view, though I’m reluctant to advocate it for obvious practical reasons if nothing else.
The tricky part is that sometimes people will hold moral beliefs which are founded on either subjective, or mistaken premises and that’s where morality potentially becomes relative.

That’s why I feel the best possible framework to have is one where we can essentially have our cake and eat it to an extent, by finding rational justifications for our morals and if same cannot be found, modifying or discarding them as appropriate. If this is successful, we can rely on objective reasons for making judgements about any culture, while also recognising that some things are indeed relative (like eating dog meat) and as such, neither the Chinese nor yourself are wrong in your ‘gut’ opinions.

Or maybe not….