I understand how hard this must be to witness, but I think it’s important to be honest about what’s going on here. The issue isn’t the other girl, it's your daughter’s response to her. That’s not something the school can realistically fix.
Pastoral teams are usually there to manage day to day concerns that affect behaviour or learning, not to provide long-term emotional support. Most pastoral people aren’t clinically trained, and with limited time and resources, they tend to work reactively. It’s not realistic to expect them to deliver what is essentially therapeutic work. If you, as her parent, don’t know how to support her with this, it’s unfair to assume the school will.
Your daughter sounds like she’s struggling with low self-esteem and possibly a fixed mindset in believing that if she can’t be “the best”, there’s no point trying. That’s serious, and if left unchecked, it can spiral. But the presence of a high-achieving peer hasn’t caused this. It’s exposed it.
You say she thinks there’s nothing wrong with how she’s thinking and this is exactly why therapy would help. Not because she’s broken or “acting out”, but because she’s got stuck in a way of thinking that’s painful and self-limiting.
If she’s already rejecting the counsellor at school, it might be worth exploring someone with a different approach or a better fit. School counselling is usually very short term anyway, and your DD needs long term counselling.
It’s difficult and it's expensive, but this is a long game. Your DD needs help to build a sense of self that isn’t based on comparison. No school can do that for her.