I am married to an anti-Catholic woman - though I have at least gotten her to think about becoming a Catholic and she is allowing me to raise our children Catholic. However, we used to fight terribly about the Church, and I've often thought carefully about this issue.
First, as you well know, you have a grave duty to your children - but also to your husband.
Your husband's activities are threatening your children's eternal souls. By depriving them of the sacraments and, worse, filling their minds with hatered towards the Church, your children will suffer grieviously - and possibly eternally. While your husband has the first responsibility for this terrible sin, why should you stand by and let him do this to them? You cannot control your husband, but you can control your own actions.
What I would do in your place is prepare yourself for battle, and it's a battle for your children's souls *- because that is what might be stake. *This process should take no more than two weeks.**
Step 1: Find and retain a good lawyer, preferably a Catholic one who knows what is at stake. The lawyer should have at least 5 years of full time family law experience. Find out your rights. Prepare yourself to file for divorce in a manner that maximizes your chances to retain custody. Do this without informing your husband. Don't get divorced yet! You are simply preparing yourself for battle so that if you do have to fall back to divorce you will be ready ahead of time.
Step 2: Learn. Research Catholic.com here, and talk to the apologists - yes, you can call them and talk to them. Find out how to answer your husband's false claims about scripture and the church. Practice how to debate, gently, with someone else on these issues.
Step 3: Confront your husband, keeping in mind that your first goal is to help him to come back to the Church. Tell him straight up, but gently, that what he is doing is endangering both his own soul and those of your children. Tell him that you will discuss his issues with him. Do everything you can to stay calm, without yelling or getting into his anger. Otherwise, you will be falling into the same sin of anger.
Attempt to get him into marriage counseling.
Think this is all too hard and difficult? Think about the suffering your children will undergo in this life, and worse yet the next. Where the stakes are eternal, the dedication in this life must be total.
Ultimately, the last card when all else fails is to tell him that while he can take risks with his own soul, you will not allow him to endanger the souls of your children. If you get to this point, you must know that he will likely reject your ultimatum and you will be in a divorce situation. Therefore, make this the last possible statement after everything else has failed. For example, if he threatens divorce first - then sidestep the issue and tell him truthfully that you don't want to get divorced but that you must resolve this issue with him. Try to avoid the ultimatum on the first or even second confrontation - remember, your goal is to get him to turn around. This will be very, very hard.
According to the Catechism of the Catholic Church, attempting to divorce is gravely wrong; however, separation may be allowed under grave circumstances. Divorce itself is impossible (though an annullment MIGHT be - but don't count on it). Therefore, be prepared to be single for the rest of your life.
Your husband permanently depriving your children of access to the sacraments and engendering hatred towards the Church is, in my opinion, grave justification for such separation. However, you should not take my word for it. Seek counseling from an orthodox priest.
If you are forced to separate, then immediately file for divorce and custody in conjunction with your lawyer. Since you are prepared, you will have an advantage. If he files for divorce first, then you will already be prepared. If you do fight, you must fight to win; accept no settlement that does not involve primary custody for you. It is highly likely the secular courts will grant you both joint custody, so he can still preach his filth, but at least if you have primary physical custody you can provide a home where love of the Church and the sacraments is strong. And you will have to do that - like, requiring a rosary a day from your kids and setting up a shrine to Mary in your house. You will need solid catechesis to keep your husband's poison from winning over the kids.
This is a terrible situation you are in. I have been in it - and to date have been blessed by my wife coming around (or at least not impeding my daugthter's access to the sacraments). However, I have been through divorce before (first marriage declared null by the Church), and I know what I'm talking about. It's very horrible and painful and I wish it for no one.
However, when the stakes are the souls of your children, sometimes you have to be willing to stand up, put yourself on the line, and fight with all your strength.
I will pray for you. Please feel free to PM me if you wish.