Lewis didn't really have a problem with Tolkien's Catholicism, except once. And it was about Lewis's wife, Joy.

So Tolkien always felt that he felt he failed a little bit by not getting Lewis to come all the way around from paganism of a stripe (see: Sprits in Bondage, which is basically WWI neopagan poetry, except neopaganism wasn't an organised religion at the time) to Catholicism. But the social currents of the day was a sort of ecumenism between Oxford Movement Catholics, which Tolkien was, and high church Anglicans, which Lewis was. Lewis rather embodied the sort of "all-English" ecumenism of the Anglican Church, which was going through a liturgical renewal alongside Catholicism in England at this time. The English Missal of 1912 drew from the 1662 Book of Common Prayer, and the 1927/28 Book of Common Prayer was influenced by English scholars visiting Catholic sites on the Continent before and after WWI. So there was this sense of exchange at the time where the theological differences weren't really that marked, and the two seemed to have more in common theologically than not. Gabriel Herbert's 1935 Liturgy and Society really shows this common ground and the kinds of dialogue happening between Anglicans and Catholics.

But then there was the matter of Joy, Lewis's fiancée. She was a Jewish divorcée, had a son by her first husback, and terminally ill. She converted to Anglicanism and moved to the UK, where she met Lewis. They got engaged. And Tolkien hit the roof. He was livid at the idea of Lewis marrying a divorced woman. It was unthinkable. He lost it. And he damn near threw away their entire friendship over the whole thing. They didn't speak for some time, because Lewis married her anyway. Tolkien felt utterly betrayed and that Lewis was doing The Worst Thing Possible, because Tolkien wasn't one for self-restraint in anything and took almost all disagreements with his friends very, very personally. Lewis, however, was undaunted and unimpressed, as was Joy, it seems. Relations were frosty, to say the least.

Tolkien got his shit together, consulted a Jesuit spiritual mentor who thwapped him about the skull with some sense and decency for a dying woman and his own best friend. Tolkien returned to Lewis to apologise, likely because it looked like Joy was about to die and that was a hell of a thing to leave unresolved. He met Joy at last, her son Jack, they all made amends, and Tolkien came to like her, by all accounts. She lived a bit longer, and Tolkien and Lewis had no further substantive theological arguments.

It's hard to piece this together, frankly -- you really need to delve into their respective letters to get a sense of this argument, and that's what I'd recommend you do in order to understand the theological discussions and disagreements they had. And understanding the Liturgical Movement, the Oxford Movement, and the ecumenical century in England is also important. Dom Alcuin Reid's The Organic Development of the Liturgy and Ernest Benjamin Koenker's The Liturgical Renaissance in the Roman Catholic Church are two good places for context. Obviously, The Letters of J. R. R. Tolkien is essential, and Colin Duriez's Tolkien and C. S. Lewis: The Gift of a Friendship is another good place to understand their relationship. I can pull up other sources on their relationship and on the context of the first half of the 20th century ecumenism in England if you want, but those are good starters for ten.