I can see that your concern is very close to your heart and I applaud you for your struggle with it! First of all, let me tell you to take a deep breath! What your Director of Religious Education has said is a truth. But it is not one that should stress you out, but one that should invigorate and challenge you to a higher goal.
Let me explain. Home is where life is full of exploration, conversation, and security for children. It is also the place where children see first hand how life is lived. What they see, hear, and model there is a clear clue as to how they will view the world. When they see the family kneel down to say night prayers, or pray at meal time, or grope around on a Sunday morning trying to give everyone a turn in the bathroom as they all get ready to go to Mass, they learn that faith is a part of life. It is the time they encounter Christ as he lives in us. (We are the face of Jesus to our children.)
So much of the faith is about living it. And that is your role. You show your children what it looks like to live as a Catholic Christian in the world. Along with that, the way you share your experiences of faith - like stories of your own First Communion, or Confirmation, etc, then they are free to imagine and ask questions. It doesn't matter whether you know the answer to a question about God. What matters is that you are open to discovering the answer with your child. Your role as one who helps your child encounter Jesus is where your biggest teaching moments come.
Because of all the time you get with your child, because of the love your child holds towards you, your moments are more important than any other. Your actions teach. Your words teach. And your love teaches. Learning to relate with Jesus is not about a lesson plan or a degree. Learning about Jesus is about learning what loving Him is.
If you don't spend time speaking about Jesus, or modelling Him, then your child won't see the importance of know Him. So you don't know the Bible. Get one that is appropriate for your child's age and read together the stories of the faith. Your child isn't going to ask you what the 7 Gifts of the Holy Spirit are. And if your child does, you can research it. Your child will ask you questions which require deep thinking and conversation. They may not have a "correct" response per say, and that's when you encourage your child to keep asking and keep praying. God will lead the right people to you when the time is right.
The role of the Religious Education program is to supplement what you do. If you work as a team with your Catechists, then your kids can grow up with a healthy view of Christianity, and be equipped to face the world when the time comes. But it takes you to lead the way in setting the atmosphere of openness and acceptance to things of faith. And that is how you are the FIRST TEACHER OF THE FAITH.
Good luck! I'll be praying for you!!