Diets, my DD is 10 and doing g6 violin in July. She changed teacher in January, as her previous teacher was similar to yours. DD went from g2 to g5 last year, not without huge complaining and digs by her teacher. After g5 she wanted to put a stop to exams for a year, so we found a new, lovely teacher who has just said DD is already g6 standard. Have to travel though.
This is a control thing. Teachers like this make you feel pushy, and in contrast, their higher goal is musicianship and a wide repertoire. The thing is, it is not some altruistic endeavour. They are being paid and, I think, should take notice of the child/parents goals.
The agreement to take an exam, moving it, then not entering him is typical of this type of thinking. And not very motivating for your son. I was a bit like you about moving DD but it has worked out very well. Her former violin teacher still teaches her piano but that has to stop now too, as she said DD was being entered for g2 spring exams, and then changed her mind because DD had a violin audition and she thought the "stress" would be too much.
When/if you approach a new teacher, do tell her from the start what your goals are. If you have a musical child, this shouldn't be a problem. You could also tell them that you are looking for a new teacher as the former one wasn't a good fit as she was very relaxed about goals. You're in central London, it shouldn't be too difficult to find a motivating teacher. Makes the world of difference.
I'm sure others will disagree, but frankly, music lessons and everything around it has a financial cost and you have not unreasonable plans, which are quite common. If she is not on board you should find someone who is.
We were at a violin workshop over the last few days, and one of mothers has a 10 year old post g8 daughter about to go to a music school. They had changed teachers several times, because, the mum said, with a dismissive wave of her hand, they didn't appreciate the bigger plan. I understand that takes a certain confidence.