Breast binding can and should be compared to other activities where an individual seeks to alleviate psychological pain by seeking physiological change.
As it frequently causes permanent damage to skin, muscles or bone, always causes physical discomfort with physical pain experienced by three quarters who bind, it can legitimately be compared to and arguably be categorised as self-harm.
Like other forms of self-harm, it is not an effective way to deal with the underlying emotional distress and any relief it provides can therefore only ever be temporary.
Although the 97% figure from the survey quoted by the OP is already bad enough, I should emphasise that 97% experience at least one negative outcome, and more than half of them experienced at least two negative outcomes.
The authors surveyed only on the 28 negative outcomes most commonly reported in the medical literature and information provided by those who bind.
Here they are in full:
rib fractures, back pain, chest pain, rib or spine changes, bad posture, shoulder pain, shoulder joint ‘popping’, muscle wasting, numbness, headache, overheating, fatigue, weakness, lightheadedness or dizziness, cough, respiratory infections, shortness of breath, heartburn, abdominal pain, digestive issues, breast changes, breast tenderness, scarring, swelling, acne, itch, skin changes and skin infections. [my emphasis]
Separately mentioned were spinal misalignment, fluid build-up in the lungs, collapsed lung, sores and permanent skin damage.
Several skin issues, which were experienced by over half of survey respondents, are associated with poorer outcomes if "chest reconstruction surgery" is pursued later. That's one of the reasons why the transmasculine community is so concerned about the safety of breast binding.
The survey also only asked adults about their experiences. In the developing child negative outcomes can be expected to be worse and occur more often.
That's because there are no safe binding methods. There are no clinical guidelines for binding safely. Commercial binders, which are perceived to be safer by the transmasculine community, are in reality associated with more negative outcomes, but even sports bras and minimisers are included in the list of binding methods leading to negative outcomes. This is not because sports bras are inherently unsafe, but because they are designed to be worn for shorter periods and not for the ten hour duration on seven-days-a-week most respondents chose to wear them.
That mother then MnaWoman is not only overstepping a boundary in allowing something that the parent of another child has expressly forbidden - something she has no right to do - she is in my view also enabling your daughter to self-harm. My guess (and hope) is that she hasn't done the research and that she means well, but is ultimately ignorant of the issues.
I would advise her that she is interfering with your parental rights without having the legal right to do so, wholly without having undertaken a risk assessment of the negative outcomes for your daughters health and development and that this interferes with your child's rights under Article 6 of the UN Convention on the Rights of the Child (UNCRC).
(This article forms one of the four main principles of the UNCRC and encompasses the right of the child to develop and survive in conditions that do not negatively impact on their physical or mental health.)
Obviously that other mother is not a state actor, local authority or a public sector organisation who could be held to account for breaching a child's rights under the UNCRC. But I would not hesitate to point out to her that every child has human rights protected under the UNCRC and that she is not only interfering with one but several of the rights your child has with her action.
The other relevant articles are concerned with your rights and responsibilities to look out for the best interests of your child and for your child having a legal guardian as a champion for their wellbeing. No random person can simply assume those rights, however well meaning.
Furthermore I would inform her that were your daughter to attend PE lessons while wearing a breast binder and collapse as a result of a punctured lung thanks to a rib fractured by the binder (this can and has happened) you would not hesitate to report her to the police for causing grievous bodily harm to your child. I would point out that she has no liability insurance and no legal indemnity if your child should be injured as a result of her buying a breast binder against your express wishes.
I have no idea whether any words to her will have any effect, but I would aim my ire squarely at that other mother and express it as coldly as possible. And rightfully so.
Then, as you probably are already doing, I would talk again to my daughter about the pitfalls of self-harming behaviours and about boundaries.