Not a good idea. It might push your child over the edge and drive a big wedge between your child, you, and your wife. This can be a very sensitive issue.
Why not? Was it the wrong counsellor? Did your child feel judged? A competent therapist, regardless of qualifications, recognises that s/he cannot work with everyone due to personality clashes or other reasons. I've run into this issue myself. The only solution is finding another counsellor. It sounds like someone qualified in gender issues would be a good idea.. theropy and stuff has not helped at all ...
Divorce and remarriage is a significant blow to a child. Both my parents were teachers, and children of divorce usually lost a year because they are too upset to concentrate on their schoolwork. Some act out, and some become reclusive.
I don't know your child, and this is purely speculation on my part, but perhaps the breast binding is an indicator that there is a underlying problem that is not necessarily gender-related. Children sometimes do some strange things when they are hurting and trying to attract attention that they have a problem.
I'm wondering if a pediatric psychiatrist can help here. A referral from your family doctor or pediatrician may be required to obtain an appointment.
Getting to the root of this issue is likely to take some time. I wish you well in getting things resolved.