You could take this to court, as it really isn't up to the mother
to determine whether to allow your husband visitation with the child based upon the child's wishes. Before doing so, I would consult with an experienced family law
attorney in your area who can give you firsthand advice and counsel as to how the judge assigned to the case is likely to rule. I am sure that you will not be surprised that some judges are very child friendly in such circumstances, and treat them as if they were miniature adults, and other judges think of them as near infants incapable of doing anything except eating and pooping. Having the advice of an attorney who knows the "players in the game" will be invaluable.
There are other factors that may be at work. The mother may be placing subtle pressure on the child to not see her father, and the child may be making the decision out of fear for the results of not making that response, or as a response to the pressure. By allowing the child to be the decision maker, she is making the child feel "grown-up", which children that age like. There may be some type of overt or subtle reward from the mother for making the "right" decision (i.e.-the decision that the mother favors).It could be that for some reason the child does not feel welcome in your home. There are other potential explanations, but they all are just guesswork on my part.
Another avenue that you could take is to ask the court to appoint a guardian ad litem to respesent your daughter's interests in the case, but it has been my experience that many GALs view themselves as spokesperson for the child's wishes, rather than investigating the matter thoroughly and reaching an independent decision as to what is best for the child, so you should be cautious.
As to "proving" that the child is not capable of making this type of decision at her age, I agree with your premise, but suggest that you be careful using this approach. I suspect that the mother will twist this around to use against you by telling the child something along the line of "I think that you are smart enough to make decisions like this but they don't." That is sure to lead to resentment if not anger on the child's part, and to long term problems. In situations like this, judges may speak to the child in private to evaluate the child without the pressure of the child having to say anything in front of the parents, and 'determine' the maturity of the child from that brief conversation, so if the child is well spoken be careful of making the child sound too much like a young child rather than a pre-teen.
I hope that this answers your questions. Please click Accept so that I may be credited for my assistance. If you have follow-up questions, please reply to this post. Thank you and good luck.