CS Lewis never really makes a case for the truth of Christianity. "Surprised by Joy" he gets very close, argues for his conversion to theism. But the transition from theism to Christianity is just sort of brushed over unfortunately.

He said in Surprised by Joy that "The Everlasting Man" is what made the Christian story make sense to him for the first time, so I've been reading that. It does seem CSL took a LOT of influence from it so far.