[Solved] Child calling step mum Mummy Mum wants enforcement

So here I am again, firstly I must say a huge thank you to Mojo for the help I have received. I seriously wouldn't have been able to do it without you.
Court is in 1 week, quite straight forward.. DHR, the second one. Position statements have been submitted and exchanged. All fairly expected.

So firstly and a problem that is niggling me.. My child came to live with me 3 months ago. Long underlying problems which finally the last straw was her trying to kill herself with my child in the house. He saw the effects of her self harming and listened as she told him she had taken x amount of tablets, she was going to die and that he would have to come and live with me anyway. I applied to the court for residency, full residency.
S7 ordered by Childrens services, the result was to be expected, she has serious problems with drink, binge drinking in particular, he also highlighted that she was dishonest but he sat on the fence and suggested a shared care agreement. I am not up for this. Anyway, the mother is suggesting that I take an undertaking in court to stop my child from calling my wife 'Mummy'.
A little background on this... It is not as straightforward as you might think...

My son wanted to call my wife Mum early last year, my wife was not for it AT ALL. She saw it that my son was comparing her to his mother and that she didn't like one bit. Anyway, this stuff happens 3 months ago and son is even more determined to call her Mummy. Wife didn't know what to say or do so insisted that he has one mummy and she was his step mum. That was it. One night son came down stairs crying to tell us that he was upset because he wants to call my wife Mummy. My wife decided to get some advice so spoke to a psycho therapist. The psycho therapist said that my son is lucky to have my wife in his life and obviously feels that she is a mother figure for him. She said that by my wife saying no to him it was rejecting him further so she should try and embrace that and think of an alternative name. Son didn't want to call her Mummy xxx so has stuck with Mummy. My wife still to this day has never once encouraged him, still tells him his Mummy is his Mum. Now his mum has said that we are encouraging this and want me to do an undertaking. She clearly feels very strongly about this. My sense is that she is jealous of the bond which her son and my wife share because it is evidently more wholesome than the relationship between them. Given the choice my son would spend a day with his step mum than his Mum. The school in which my son attends even refer to his Mum as my wife and not his Mummy. His Mum has never been to the school, never intimated making a relationship or even making herself known to them. She has no interest in his education, never has done. My son who is quite wise for his years sat down with me and my wife the other day and said "I wish I could go back to being a baby so that you could be my real Mum". This broke my wife heart, so touching but so sad at the same time. My son has never been told to call my wife anything other than her name. It was him that decided upon this but the mother won't have it. She can't see the damage that she has done, the sense of insecurity she has created for him which has led to him doing this. Mother has no problem with son referring to my other child as brother nor my wife mum as Nanny. She just comes across as very jealous and although usually I wouldn't go along with this it is very strange circumstances and strained for my son and my wife who has done absolutely everything possible for my son which includes full time care for him the last 3 months without so much as moaning once! Dentist and doctors appointments have been strained because she has no parental responsibility, something else that we would require should I get full residency.

What is your stance on this? Are we wrong? Have we taken the wrong advice?
How would I address this in court ? It is bothering my wife tremendously as she has only ever done what my son has wanted her to and that is love him.

Thanks in advance 🙂