This could conceivably be done by creating a notion of special-interest delegates. The groups they represent could register, be voted on by the people (general election). If they received a sufficient vote (10-20%?) then they'd exist for some period of time (10-20 years?). People could then register with the special-interest groups of their choice (perhaps a limited number, with a minimum period of membership to avoid people attempting to spoil elections, they'd have to choose to join groups they cared about or groups they wished to spoil, but have a hard time doing both). Delegates would be nominated by various participants within the special-interest group. Consider that, a labor delegate, could be nominated by various unions, as an example. Or in your Catholic example, the Knights of Columbus and the Franciscan Order.
Delegates would be a sort of elected lobbyist, but instead of operating behind closed doors (so much), they'd have to layout their position in campaigns before being elected to such a position.
I'm not sure this'd be a good idea overall, it has a lot of downsides that I don't have time to describe as I'm about to head out the door. But there's some merit to the concept.
Oh if it wasn't clear, I was being facetious. This is an awful idea. All the groups mentioned above already have a legal, constitutional right to redress the government, being groups composed of individual citizens. Lobbyists, paid or not, elected or not, do not deserve floor time in Congress. They currently do not enjoy this privilege, thank goodness.
Delegates would be a sort of elected lobbyist, but instead of operating behind closed doors (so much), they'd have to layout their position in campaigns before being elected to such a position.
I'm not sure this'd be a good idea overall, it has a lot of downsides that I don't have time to describe as I'm about to head out the door. But there's some merit to the concept.