If you feel confident you can afford it until she has taken her GCSEs, then I would go with JAGS. Switching to state at Sixth Form is not uncommon for independent pupils and it can work out very well.
As for the grammars, well you know your DD best and whether she is likely to thrive in a fairly intense atmosphere, and whether you, as parents, can act as a buffer against the pressure when needed.
I’m hesitating to say this because your DD is clearly bright and has worked hard to be holding those places, but I had a DD at one of the schools you mention, and have a friend with a DD who was at another of them, and what they have said makes me wonder if they are the best fit for a “sensitive, introverted” child.
This is just anecdotal, but DD said she didn’t know anyone who had been to her school “who didn’t end up a little bit mad”. There certainly seemed to be an alarming amount of self-harm, eating disorders, anxiety and depression among her classmates.
DD was happy there but she joined in Sixth Form, so she hadn’t grown up in that pressured environment. Someone she knew who had, later said that overall she would give the school a ‘B’ “because ‘B’ was never quite good enough for them.”
My friend’s DD suffered from anxiety, and to her school’s credit, counselling was suggested. When my friend agreed but asked if the sessions could be discreet because the DD would hate to feel singled out, the school replied in words to the effect that it would be more abnormal if she wasn’t receiving counselling.
It’s a tough decision. But a previous poster does make a very good point in saying if you take the grammar now you can always pull her out at any point.
Best of luck whatever you decide.