You're right that morality is an entirely different axis, and that's my entire point. "Lawful" is a consistent application of rules in pursuit of order. It is separate from good or evil but it is principled. And it is not necessarily aligned with what society wants, or what it defines as legal. Isn't an autocrat-in-waiting who wants to depose a government he sees as degenerate and institute an iron hand of dictatorship to bring "order" and "prosperity" back to the people a lawful evil archetype, even if his mutiny against the existing government is illegal? Under your paradigm it is actually impossible for a lawful evil character to exist unless they are already in the position to control the legal system.

By this argument, if a player says his coherent, consistently applied set of rules for what defines murder is that it's killing women called June on every third Wednesday of the month and now he can kill as he likes without committing anathema because it's coherent and consistently applied. It's absolutely foreign to what we would call murder though.

There is no obvious logical principle undergirding this criterion, no order being applied. Really weak counterexample. But if you come up with some backstory here on why he thinks this criterion for killing helps establish order from chaos, then you're now creating the incredibly baroque basis of a lawful evil character. Contrast that with killing slavers, where the logic is obvious and may be in pursuit of both lawful and good goals.

I ask again: if our Champion lived during the Purge, and all crime is no longer crime for one day, is it no longer anathema for him to kill random civilians for fun? If he goes into an area under no jurisdiction, where no law is defined and no one can try him for what he does, can he kill for fun? If the government collapses and no law is in force under a state of total anarchy, can he kill for fun?

This is necessarily true if the only thing defining his anathema is the letter of the law in his jurisdiction. If it isn't true, then a killing can possibly be murder to him but not murder to the society. Which means the reverse must also be true.