This thread is not about a "solution to a child who insists they are not the gender they have been designated at birth" but about children who, specifically, are being facilitated by a chain of bath-and-beauty retailers to buy breast binders and collect them without the knowledge of their parents.
The solutions to that situation depend upon a complex set of data - each child will be different and their situation will need different handling. Plus, of course, if the parents are unaware that their daughter has bought one, they are not able to consider what the motive is, or to take any action to handle the situation at all..
I have not heard of Lush imposing any age limit for this purchase, so I assume they will sell to anyone who produces £7.
The pre-teen girl who is one of the first in her class to develop breasts and finds every boy in the playground is fixated on staring at them may think that strapping them tightly down will save her embarrassment. She needs to be assured that most of the other girls will soon catch up and share the limelight, and the novelty value will diminish. Meanwhile, she needs to be bought a comfortable bra of the right size to support, constrain and restrain them without crushing the life out of them, and to be encouraged to stand up straight, move naturally, and not be ashamed of her shape.
The girl a little older may be aware that she is entering into a new era in her life, when she must add to her relationships with family and friends the responsibility for her sex life and for how she is seen sexually and interacts sexually with others, and she may be overwhelmed (consciously or subconsciously) by the magnitude of that looming reponsibility. Binding down her breasts may seem the solution to their dominance over her life. She needs guiding through the changes happening in her life.
Younsters are copycats at heart. They complain about having to wear a school uniform, but if the in garment is a certain make or colour of clothes, or length of skirt or presence or absence of trouser turnups or back pockets, then the cry is, "But everybody has one!" There will be some girls who buy a breast binder because Tracy across the road has one, and it is unthinkable to not be upsides with Tracy, or perhaps Tracy hasn't got one, so here is a chance to be one ahead of her. Treatment of this motive depends on how indulgent parents are to Tracy-envy, but they should be pointing out that it is not a good reason to waste money.
The late teenager or early twenties young person who is certain of their gender and/or sexuality and their place in the mosaic of adult relationships is probably coping with the inevitable problems they meet. They are also probably quite rare.
Then we come to the girl who genuinely wishes to look, act and feel as though she were not a girl. Even this is not a single category, because the fourth dimension of time is involved. Many girls wish at some time that they were boys, for various reasons. Boys still appear to them, after all the efforts of feminists, to have a freer life. They don't have messy periods and suffer from cramps every month, they are less likely to get pinned against a wall by a randy acquaintance, they still get the more interesting jobs and are paid more for them, they are not expected to be gentle and helpful and refrain from using bad language or fists when provoked. If drastic action is undertaken to transition and then regretted at maturity, it is too late to reverse all the changes. If young girls are seriously contemplating the chain of changes which begin with binding their breats, then either their parents should be aware that they are embarking on this and be considering whether a professional opinion is needed, or they should (of their own choice or by advice by an expert) be under medical supervision by someone who can look deeply into their feelings and motives, and guide them through the maze.
None of these girls should be getting advice from a sales operative for a cosmetic firm.