I don't have much advice to give but I will pray for you. I don't know you as I have not been on this bored long but I know about having babies.
The thing with babies and women is that NO MAN will ever understand what having a baby does to a women. I got so sick in my pregnancy that I got to point where I cried out and I could not imagine purgatory being worse. I actually developed an allergy to my baby and had burning welts all over my body. The only thing that kept me sane was knowing it would only be 9 months of h*** and not a lifetime. Plus after the baby comes you literally don't sleep for a year and your hormones are all screwed and it is hard. No man will ever understand. I have an amazing supportive husband and he still has no clue. So the best thing you can do is show empathy for the vocation she has and all she does already and try to help out more. Ask her what scares her about more children and ask her what she needs from you to get over the fear. She says she is scared of getting sick and tell her if she does you will take care of her or hire help to care for the children. She probably is feeling alone. Being a mom can be lonely in a way that when a women becomes a mom 99% of her life changes but for a man only like 10% so he will never understand how demanding motherhood is. Don't blame her or lecture her about God. Just say you understand why she wouldn't want more because it is hard but she is a great mom and wife and her babies are so lucky to have her. Tell her you can't stand the thought of not being with her in heaven and you will do whatever it takes to help her get there . I have tons of friends and the main reason they don't want more kids is because they don't get help and feel alone and sleep (sleep is the biggy.....offer to watch the kids so she can nap at least once a weekend). Start with the little things and it might help to move the big things.
Call home from work to see if she needs help or you to pick up anything. Come home early and tell her you will watch the kids for an hour so she can have a break, Etc.
You get more with honey than vinegar with women so show EMPATHY and be there. Work on being a supportive husband and make sure you tell her you love her and appreciate her. So many men have problems with words but it is worth it. Ask her to pray with you about it.
Most importantly LOVE her and make sure she knows it. Don't give up.