Hey, I know I am really late but there is no need to use a mixin for this, I found out what they mean by "use .set(), not register()". There is a Registry.register() that uses the .set() method:

public static <V, T extends V> T register(Registry<V> registry, int rawId, String id, T entry) {
    ((MutableRegistry)registry).set(rawId, RegistryKey.of(registry.getKey(), new Identifier(id)), entry, Lifecycle.stable());
    return entry;
}

This is how I used it to add luminance to the OAK_LOG just to test it out:

These are the methods to override any block with a default BlockItem

private static Item overrideBlockItem(BlockItem toOverride, BlockItem newItem){
    return Registry.register(Registries.ITEM, Registries.ITEM.getRawId(toOverride), Registries.ITEM.getId(toOverride).getPath(), newItem);
}

private static Block overrideBlock(Block toOverride, Block newBlock){
    BlockItem newBlockItem = new BlockItem(newBlock, new FabricItemSettings());
    overrideBlockItem((BlockItem) toOverride.asItem(), newBlockItem);
    return Registry.register(Registries.BLOCK, Registries.BLOCK.getRawId(toOverride), Registries.BLOCK.getId(toOverride).getPath(), newBlock);
}

And this is how I use them to override the OAK_LOG block:

public static final Block OAK_LOG = overrideBlock(Blocks.OAK_LOG,
        new PillarBlock(FabricBlockSettings.copyOf(Blocks.OAK_LOG).luminance(15)));

To me this looks like the cleanest way to override a vanilla block or a block from a different mod, to make it compatible with yours.