Jen, prayer, prayer, prayer. Havard's suggestions are very good. A great approach, but it may only delay his adamant opposition. In all organizations there are rules. The Catholic Church is no exception to that. Sometimes when a rule isn't met well by a non-Catholic, they object to the whole church and not simply that particular rule.
I have a similar experience with my wife. I have been blessed to be married to a WONDERFUL woman for over 30 years. She is Protestant (Church of Christ). We have twin sons, who are both adults now. They were not baptized as infants due to her opposition. Regardless of my pleading, begging, etc., she just couldn't support it. To make a long story short, one of the twins is now Catholic, one attends the Church of Christ. :)
So what am I doing about it now? Praying...in private. This is the first time I have told anyone this. About a year ago, I was doing some reading and came across some information on the green scapular. I bought one and placed it in a drawer in our bedroom. Simultaneous to that, I began praying daily the prayer, "Immaculate Heart of Mary, pray for us sinners now and at the hour of our death." I then pray to Mary to intercede for my wife. I do this daily. Every day. In silence.
You know what? Just the other day, my wife broke her bible open and tried to make an argument against a Catholic doctrine concerning...MARY! I defended the position and the "discussion" began. She stated that she's just not sure what church to believe any more. She said there are definitely things in the Catholic church she cannot ever accept. She also said there are things at the Church of Christ that are beginning to bother her. But she stated emphatically, "I will NEVER become Catholic!!!"
Inside, I was praying to Mary the whole time. When the conversation was over, I thanked Mary for interceding on my wife's behalf. The way I see it, the Holy Spirit is working in my wife's life in ways none of us can understand. He is getting her to think about her faith and to question that from within. Even with her stance, I feel optimistic that one day, ONE DAY, she might convert. I feel this way because 1-God can do anything, 2-My wife is beginning to question her own faith tradition, and 3-There are thousands of people who at one time held even harsher positions and they are now some of the best defenders of the Catholic Church (Scott Hahn comes to mind)!
So, Jen, have the conversation but remember 1 Peter 3:15-16:
Always be ready to give an explanation to anyone who asks you for a reason for your hope, but do it with gentleness and reverence, keeping your conscience clear, so that, when you are maligned, those who defame your good conduct in Christ may themselves be put to shame.
Be patient. Be kind. He will say some hateful, hurtful things about the holy mother Church. Let him. Remember that Jesus told us that all his disciples will be persecuted because of him. Be firm in your position. Try to understand his position, even though you will feel he isn't even considering yours. If he will listen, explain how important your faith is to you - even though mere words cannot ever express these feelings adequately. Try to make him see that this is not a group to which you simply belong. It is a part of you. To criticize the church is to criticize you.
It might help to ask him to give you three points of discussion and only three. They need to be as specific as possible. If one could be taken several ways, try to lead him to define which way he wants to discuss it. Then limit this discussion to those three points ONLY. Don't let the conversation creep into other areas. Limit them to what was agreed upon. If he brings up something out of scope, tell him you will be happy to talk about that point later but for now let's finish this one topic. Ask him to explain his position. Then tell him you understand how he feels. That will make him feel less defensive. Tell him that if you were in his shoes, you might feel the same way. That, too, should help him lower his shields. Then you can kindly explain the Catholic position. You can even begin with the caveat that you don't expect him to accept or even understand the teaching, but please try to understand why this is important to you. With that, he can maintain his objection while you at least get to explain your position.
Your objective here is to plant seeds. You never know which seeds will die and which will take root. It may take some time for the rooted ones to begin to grow, but keep planting. He may object with great enthusiasm at first, but after contemplation he may soften his position.
The main thing is to be kind and not argumentative. If he raises his voice, soften yours. If he gets animated, get even calmer. Ask that the discussion remain a discussion and not an argument. If that becomes impossible, tell him in the most kind way that you cannot continue until he is able to control his emotions and talk calmly. It may take a few nights or weeks, but it is important to have these discussions.
And remember. You don't have to win every discussion. Your job is to plant the seeds, pray for him, and let the Holy Spirit work in his life. Remember the words of Archbishop Sheen, "Win an argument and you lose a soul."