It is unacceptable for a mother-in-law to be verbally abusive to her daughter-in-law. Verbal abuse, jealousy and hatred to that extent can be part of a personality disorder. I can recommend researching in detail the characteristics of the malignant, narcissistic woman and mother. In insulting someone you love, she is also abusing you. Mothers with a narcissistic personality disorder (as opposed to just being self-centred) are competitive in terms of who their child loves most.
How your children see you behave with your mother and wife will affect their own marriages down the line. Can you ask your mother not to verbally abuse your wife? If you feel you can't, that is revealing something about how your mother has trained you to relate to her, i.e. to remain passive and take the abuse; to not have enough self-worth to even politely ask her not to be verbally abusive.
Your duty is to love and protect your wife. You are the head of your household. You also have to protect the well being of your children. If they see their mother being abused their well being will be affected too.
According to Ayatollah Dastghayb-Shirazi, you are entitled to minimise or even cut ties with family members whose bad behaviour you can't reform, or whose bad behaviour gets worse by your presence, or whose bad behaviour you indirectly condone by co-operating with them. Being good to your parents does not mean condoning behaviour that could destroy your family.