The deeper Catholic conception, or traditional Roman Catholic conception, is a concept in conjunction with and in contrast to the deeper Protestant conception. The deeper Catholic conception is the notion that when Adam was created as the image of God, he was made like God, in that he had intellect and will, but he was ontologically and ethically under-proportioned to participate in the essence of God.
In addition to God creating him in the image of God and giving him the natural gifts of reason, freedom, and will, God also infused supernatural qualities in a donum superadditum, a super-added grace. That super-added grace begins the process of ontologically and ethically re-proportioning Adam to an ascending participation in the essence of God, the end of which is no longer, Adam knowing God indirectly, through created media but participating in and seeing directly with his intellect, the essence of God.
That deeper Catholic conception—at least in part—is what Westminster Confession of Faith 26:3 forbids. It says that we have union and communion with the person of Christ but in no wise partake of the divine substance. That is a blasphemous and impious idea.
That deeper Catholic conception is the programmatic, eschatological alternative to the deeper Protestant conception. The two are comprehensively distinct accounts of the God-world relation, what the creature needs in creation, and what the end of the creature is given in beatitude.
For the deeper Protestant conception, it is union and communion with Trinitarian persons after the fall in union with Christ. For the deeper Catholic conception, it is being ontologically re-proportioned and elevated above human nature to see directly and participate in the essence of God. Bavinck calls that deeper Catholic conception “a melting union.” This is partly why the antithesis is so sharp between the deeper Protestant and the deeper Catholic conception.