If you can come up with ways to make her feel in control as much as possible - both in the decision, and as it is happening - as well as helping her find the motivation to want to do it I suspect it will help enormously. Having something nice planned immediately afterwards can help as a motivator, if it's of her choosing (not imposed).
Or a strategy some people use in the moment when they're panicking about a procedure is to focus on how they will feel afterwards not how they feel right this second - eg how proud they'll feel of themselves for doing something scary, how they won't have to keep worrying anymore because that's it done for life, how relieved they'll be....
I don't know what happened the second time, but clearly it was distressing enough for you to use the word traumatic. Generally, trauma is about feeling totally out of control and unsafe, so the best response is to help that person find some control again over what is happening to them. And to help them find a sense of safety again.
Teenagers do tend to try to psych each other out when injections take place at school, so I doubt the environment helped.
Could you book an appointment at GP just to discuss it and how they could support her with it? So she'd be attending knowing she wouldn't be having the injection and the decision would still be hers, it would just be an information gathering exercise. Then put it back to her having been there to think about what would make it easier for her (eg taking something reassuring to hold, lying down, you there/not there...)
They're all strategies that will help in her wider life anyway.