Question

is it radically true that different cultures have radically different moral codes

Answer 1

It is not a sound argument because the conclusion is not based on the premise (in other words, it is not valid). The fact that people have different views on what is moral does not mean that morality is historically subjective. This is easy to see if we find an equivalent claim

We are inclined to conclude that many cultures have actually been wrong on a variety of issues. Medieval Europeans assumed that the earth was square, that the planets were small spheres in perfect circles, and that the sun was stationary. We are now denying all of these arguments on the basis of well-founded scientific theory. The bottom line is that pure disagreement does not mean relativism. This is not to suggest that we have shown that cultural relativism is wrong. The point at issue here is that the Cultural Differences Claim does not create cultural relativism.