Your daughter does not have a problem:
1. Reports from school are that she is a popular, happy child
2. She wakes up singing and is generally happy
3. DD doesn't suffer from anxiety, never seems to be stressed about the social aspect of school.
Here's the problem:
I feel that she is missing out on a balanced childhood.
Last year I bumped in to one of her friends from school and invited her over
I'm always conscious that she needs to be entertained as otherwise she'd do nothing.
I wanted to bring her to a psychologist to see if there was something that we could do to help but DH isn't in favour.
I just think that as parents its our job to try and raise a child who is as fully rounded as possible and that we are letting her down in some way by not pushing her.
You say that your mother was controlling so you try to be the opposite. It hasn't worked. The apple has fallen very near to the tree. You may couch your argument in reasonable terms and be articulate .... but you are still controlling.
It is great that your DD is so happy and has friends at school. She is 12, old enough to have an idea of what a psychologist is and resourceful enough to Google it. If you take her to see one I think you can pretty much guarantee it will be detrimental to her happiness. She won't be singing every morning when she thinks people think there is something wrong with her.
I think you could see one but for you, to try and work out why you feel the need to make your DD into someone you want her to be rather than the happy girl she currently is.