C.S. Lewis didn’t have anything to say about the kind of complementarianism that is being promoted by CBMW among others, according to which men and women are allocated complementary, but allegedly equal roles in the family and in the church – and it is men who decide which these roles are. That is because the concept had not been invented when he died.
But Lewis did have an idea of what it meant to speak of complementary roles where the allocation of these roles is all done by one side. In the first chapter of The Last Battle he writes of how the ape Shift, a clear figure of evil in his story, and the donkey Puzzle
both said they were friends, but from the way things went on you might have thought that Puzzle was more like Shift’s servant than his friend. He did all the work. … Puzzle never complained, because he knew that Shift was far cleverer than himself and he thought it was very kind of Shift to be friends with him at all. (p.7 of my Puffin edition)
After getting the reluctant Puzzle to fish a lion’s skin out of Caldron Pool, Shift says:
You know you’re no good at thinking, Puzzle, so why don’t you let me do your thinking for you? Why don’t you treat me as I treat you? I don’t think I can do everything. I know you’re better at some things than I am. That’s why I let you go into the Pool; I knew you’d do it better than me. But why can’t I have my turn when it comes to something I can do and you can’t? Am I never allowed to do anything? Do be fair. Turn and turn about. (p.12, emphasis as in the original)
With arguments like these Shift asserts his leadership over the poor Puzzle and exploits him as his servant, to do all the dirty jobs while Shift reserves for himself all the nice ones. These arguments sound remarkably like the ones which complementarians use to justify men getting all the desirable roles in church and in family, while all the ones which the men don’t want end up being given to women.
Now Shift probably was cleverer than Puzzle, so he could justify being the one who did the thinking – although not the evil he brought from it. But there is plenty of proof that men are no better at thinking or at leading than women are, and so no justification for men allocating to themselves all the leadership roles and any other tasks that they take a fancy to.