We need not be slaves to our unhealthy growing up environment. An aware person is capable of bringing in the required changes. Since you are wiser and aware, you could stay out of the conflict occurring in the presence of the children and tell her firmly that you will not fight with her in their presence. Keeping your calm is of utmost importance. Couple/marital therapy is ideal for your case. If she's not ready, then you can do individual sessions and know how to  manage such tricky situations with ease.

Please consult a family counsellor or Behavior therapist

Irrespective of the environment that she grew up, if you both have an understanding in the interest of your own relationship and of the children,  you can bring in humongous change.  Agreeing, disagreeing,  arguing will be part and parcel of any relationship.  But how you resolve and come to an understanding is essential. Please go for a family/marital  counselling in your city.