The CCC says in §1445:
Reconciliation with the Church is inseparable from reconciliation with God.
Reconciliation with God and with the Church go together. You can’t have one without the other. Why is that? The answer, I think, is to be found in the Incarnation, and particularly in the doctrine that the Church is the Body of Christ. There is no controversy associated with the idea that the Church is Christ’s Body; it is repeatedly proclaimed to be true many times in the New Testament, as for example in Ephesians 5:29-30:
A man never hates his own body, but he feeds it and looks after it; and that is the way Christ treats the Church, because we are parts of his Body. (NJB; emphasis added)
This being the case, it ought to be obvious why exactly reconciliation with the one cannot be separated from reconciliation with the other: In Christ, the Son of God has taken on a human nature forever. Consequently reconciliation and union with the one necessitates the same with the other.
This seems relatively uncontroversial for anyone who accepts the orthodox dogmas of the Incarnation. But it becomes a bit more interesting when we ask exactly how one is reconciled with God and the Church? Jesus tells us:
If you forgive anyone’s sins, they are forgiven; if you retain anyone’s sins, they are retained. (John 20:23, NJB)
In short, reconciliation with both God and the Church is accomplished through one and the same act: the priestly sacrament of penance, confession or reconciliation. But this fact demands a follow up: who is empowered to perform the sacrament? The same passage from John’s gospel answers this question as well: the priests of Christ’s Church have the authority from Him to perform this sacrament.
If this is the case then it goes without saying that Christians and those who want to be Christian must be able to identify His Church, and this implies that His Church must be visible. All this is a long way of saying that reconciliation with Christ and His Church — as the Catechism says are mutually necessary — makes the Protestant doctrine of the invisible church rationally impossible. Why? Because if the Christian (or prospective Christian) is unable to identify where the Church actually is, he has no certain access to the reconciliation that Christ promises in John 20. The Protestant may object that he may confess to God directly, but that act only accomplishes half of what the Church says is necessary and half of what seems logically required by the Incarnation: reconciliation with the Church is no less necessary than reconciliation with God. They go together. But the idea of the invisible church offers the Christian no certainty about that reconciliation. This being the case, it seems clear that the Protestant doctrine does not pass the smell test.