Hi - I am afraid I don't really have any answers but wanted to bump your post, and also sympathise - my DD in Y5 (DC2) is like this, and I also want to try and 'scaffold' her confidence before secondary school in a year's time.
If school helped in the past, might they be able to help again? And maybe you could go and see the GP - they might be able to reassure you and your DD?
As I said, I am struggling with a similar thing here, and don't know what I am doing, but thought I'd just share what I am hoping is working for us:
I have bought a few books to go through with her: Smart girls guide to... (and insert issue of choice, e.g. knowing what to say, boys, confidence, etc.) They are American, so sometime slightly different culturally, but it's a focus for us to talk about these things without DD feeling I am lecturing her. And I think also reassuring to her that lots of girls can feel like this, and that's ok - it's how you deal with it.
I have also adopted the word 'catastrophising' and after letting her vent (e.g. violin grade 1 exam this morning) - what if I forget, what if I am nervous, what if a string breaks, what if what if what if - I say gently, 'I think you might be catastrophising', and we talk through what she'd do if any one of these unlikely events happen. I think is helping a bit, she doesn't get sucked into such an emotional, teary vortex as often.
We also found a couple of things on the internet I have printed out and stuck up in the kitchen: one a list of 'losers' such as Einstein, Steve Jobs, Walt Disney, JK Rowling (failed exams, got fired, told he had no imagination, Harry Potter rejected by lots of publishers). The other is a list of what you say, next to a list of what you could say instead: "I can't do this, it's too difficult" becomes "this is quite tricky, but if I keep at it, I'll get better". Again, once she starts, I just refer her to the list, so it is not an annoying mum pep-talk but true-fact golden advice from the all-knowing internet.
Lastly, we talk about how you have to master the thoughts and feelings in your head: acknowledge them, they are valid, but they can't affect what you want to do. So if you feel nervous, think 'I feel nervous, and no wonder, it's my first violin exam ever'. But then as soon as the negative "i'll mess up" type thought start, you have to park them, think of something else/ do something to take your mind off the issue. (Oh gosh, is this avoidance?)
As I said, I am doing this by the seat of my pants, who knows if it is working. It's a new thing for me: my son (DC1) is quieter and shyer, but very 'stoic' if that is the right word. He had his first music exam yesterday but you'd never guess unless you knew him, he just goes even quieter until it is all over.
I hope you find an approach for you and daughter that works.