What are the Comps like where you are? Are they genuine Comps with the full range of ability (unlikely in a grammar area) and do they have a decent number of bright children and set them for a number of subjects from fairly early on?
The problem with 11+ is its a one-time -only experience. If you choose not to do it, you've missed the boat and won't be able to change your mind.
Most children feel a bit anxious or pretty anxious about it. They aren't daft and realise that it determines their future school - that would make anyone feel a bit anxious and most children will say they don't want to see it the paper at some point. Whilst doing even basic familiarisation, most children will feel a bit over-whelmed at some point and think it's beyond them. It's all entirely normal. However a child who is having serious mental health issues probably shouldn't have this added pressure. Only you can judge if she is feeling the normal anxiety or there are more serious underlying issues.
If it's the normal anxiety, some gentle coaxing and encouraging and a low level familiarisation plan for the summer, so she knows you can help her prepare and have a plan which isn stressful, is definitely worth trying. If you think and her school think she's academically up to it, you might always wonder otherwise. If she doesn't pass, it doesn't matter and you choose another school....but at least you know.
Don't be influenced by the people who say it's impossible without 2 years high intensity tutoring. If she's a bright girl, unless it is a super selective of the Henrietta Barnet type, every year, students who've had a bit of familiarisation and are bright get in.
You will need to do familiarisation. No one should sit down to do verbal and non verbal reasoning in an exam without having seen those kind of Qs before. Your job is to encourage, make a plan so she is prepared in a low key way. You will have to accept it won't all be a smooth ride until the exam. Everyone preparing over the summer will find some struggles along the way...unfortunately it's par for the course and not pleasant.