Catbirds won't allow other bird types to nest in our shrubs. Under no circumstances do I advocate killing, trapping, poisoning or harming birds of any species. However, are there ways that you know of to discourage the attacking catbird?
To better enable cardinals to nest.
What about trapping and re-locating them away from your area?
I've been thinking about how to answer this and I do not know if there is really an answer. We've never had Catbirds, but Grackles that have booted Robins and Thrashers out of the trees, so they can claim the nesting spots. I never despised those Grackles as much as I did then. The preferred birds were persistent and finally found spots where they were left alone and we had young coming to our feeders and bath.
I am curious, are the Catbirds claiming just one shrub or several? Cardinals will nest in taller evergreens, but not usually very high up.
Trapping native birds I believe is illegal unless you have a permit, even if they are released elsewhere. You can verify this with your state's DNR. Maybe they can give you some advice in your situation. They were helpful to me, at least here in Iowa, when I had questions for them.
In searching I've found these:
Spray your yard regularly with insecticide so bugs don't attract cat birds. Drape bird netting over berry bushes and plants.
I don't like insecticide as many birds like insects.
Eliminate by thinning overgrown vegetation and thickets that offer the dense cover they prefer. While cat birds do sometimes nest in tall trees, they are more comfortable near the ground.
Get a roll of iridescent foil flash tape at your garden supply store. Cut it into 2 foot strips and tie them 4 feet apart wherever you want to keep cat birds away. The sun's reflection on the tape--and the snapping sound the tape makes in the win--will frighten the birds.
I think this would frighten other birds as well.
Get a liquid bird repellent containing either methyl anthranilate (MA) or dimethyl anthranilate(DMA), according to the University of Colorado Extension. Mix and spray it according to the manufacturer's instructions on all plants you want to protect from cat birds. DMA is a natural compound found in grape skins and birds dislike its taste.
I'm not sure you have ever properly identified the actual species of bird that you think is causing any problem.
Till you do that (like, by posting a picture here so that bird experts can help with the ID), all this speculation is not going to help at all.
I don't know that I'd try to manage this, but just let nature do its thing like it has for eons. Then enjoy the catbirds in your yard. If they are nesting in your yard, then they aren't likely to be harassing cardinals somewhere else.