This answer is my personal opinion, no more, but it is the personal opinion of an aging man who has read almost every word Lewis ever wrote, most more than once.
When Lewis returned to the faith, he was doing what many do, looking for a spiritual home that was also a social orthodoxy. This can be seen if you read his memoirs, Surprised by Joy. Shortly after he reconverted to mere theism, and before he was a believing Christian, he began to attend church. This suggests that he was not merely searching for a credible metaphysic, but also a community in which it was believed. In middle class England, where he worked for most of his adult life, that orthodoxy had to be Anglicanism. Moreover, Lewis was brought up in the Anglican church of Ireland. He was returning to something he knew.
The Protestant Catholic distinction is no longer a heavily political distinction in England, but it once was. Remember, the battle of Marston Moor was fought between Royalists (overwhelmingly Catholic) and Parliamentarians (overwhelmingly Anglican Protestant). Remember, when Newmann and Manning migrated from Protestant to Catholic in the middle of the 19th century there were many disapproving comments and raised eyebrows.
My last paragraph is, on reflection, over-simplified — but it gets to one of the points I'm trying to make. If you read the Wikipedia article on the Elizabethan Religious Settlement it becomes very clear that the creation of a separate Christian denomination called the English Church or Anglicanism was a highly political business in which both houses of parliament were involved. We now talk as though the differences between Anglicans and Catholics were differences of metaphysical doctrine. This is not quite so; but it is what happens to the church in political squabbles. When the squabbles are resolved, subsequent generations are left with differences of doctrine which they argue about without paying attention to their political origin. This overlooking animates both the question we are trying to answer and Lewis' own remarks in Christian Reunion. Right down to Lewis' own youth, even though the Catholic Church was and is an essential part of the Religious Physiognomy of Britain, to be orthodox was to be Anglican. If you are brought up orthodox and involved in certain professions, you don't think about orthodoxy, you 'roll with it'.
I have often speculated that if Lewis had survived much longer, sound in mind, wind and limb, he would have considered, at least, becoming a Catholic. (If memory does not deceive, his mate Walter Hooper did migrate to Catholicism.) I doubt, however, whether he would have given much attention to the political side of his choice. Politics was not his favorite aspect of human life.