The BAGA badges are a recreational award scheme. I have all six somewhere from my youth. They are not really intended for a gymnast who is competitive stream material. To be having those as the gold standard for a 4 year old is a bit odd. Good that she is going into pre-team where they will focus more on the quality of very, very basic skills than getting skills.
If you look at videos of very young competitive gymnasts in the UK who are successful you will see that the skills they perform are very, very finely tuned. That is what competitive stream coaches do. In a BAGA recreational classes the coaches are not as picky and get their gymnasts doing skills much sooner, though technically less perfect.
Once she is in pre-team you will see that she will get skills more slowly, but once she gets them they will be a lot nicer than in the BAGA setting. This is a very good thing.
SO my advice, like that of many others, is to just let her be at home. She can have fun playing around, but try to avoid "coaching" skills as they are a bugger to reteach later and that really slows down progress and confuses the child when mum says one thing and the coach something else.