ICanWearMyBoobsUpOrDown some years ago a mother in your position came to seek help from the regulars on Feminism Chat. She stayed posting regularly for many months.
We found her resources to best support her DC. Information about the long term effects of the medications, about detransitioners and other stuff she found useful whether for herself or to share with her DC
She was determined not to lie to her child that sex changes were possible and to steer well clear of puberty blockers and hormones. She'd looked into them and was sure they were not in her DC's interests.
She had joined Mermaids as the obvious thing to do but was horrified by what she found. The other parents spent meetings sharing details of how to obtain medication from websites or unscrupulous private doctors. When she left she was pursued by other members saying that her DC would commit suicide and it would be her own fault.
She shared this on MN only for several Mermaids members to sign up specifically to pursue her onto the board. Their behaviour made it clear that she was not exaggerating the issues she had with the group.
She too was paying out money she couldn't afford for private therapy. She was very angry at the weaponizing of very dodgy suicide stats by Mermaids and the misrepresenting of puberty blockers as a harmless pause.
There are a range of motivations behind an expressed desire to be the opposite sex, ranging from autism spectrum disorder, PTSD, internalised homophobia and misogyny, abuse of various sorts and family issues. You sometimes get a child expressing dysphoria as a reaction to the death of an opposite sex sibling. And, sadly, sometimes parents would prefer a trans daughter to a gay son.
These days MH professionals are being taught that any exploration of the other factors that might be driving gender issues is considered conversion therapy and is a BAD THING. I knew this was happening but actually met a woman last week whose DD, a trainee psychiatrist, had been very disturbed by this while working for CAMHS. So you're probably doing the right thing by going private. You may want to make sure the therapist is looking at all the potential issues. For instance I know being a lesbian is very uncool now, but become a transman and you're suddenly extremely popular..