Hi Gadnitt, and welcome!
I have removed references to your daughter's score from your post because score is not relevant. Any score over 121 is deemed qualified.
She wasn't in catchment for the first school - their radius is 9 miles, and we are 12 miles away. She is (we hope) on the waiting list for the first choice.
Have you told Bucks CC Admissions that you want to put her name on the waiting list?
If it is RLS that you applied to, the distance is unlikely to increase by 3 miles over subsequent allocation rounds. If it is Burnham, there is a small possibility of the waiting list delivering a place.
The question is whether anything else will take her case forward: her class teacher suggested we call the Grammar School and ask them to review her profile. I can see that if every hopeful parent did that, Admissions would be up in arms!
It will get you absolutely nowhere.
You need to lodge a formal oversubscription appeal. There is general advice about that here: http://www.elevenplusexams.co.uk/appeal ... bed-school
and information specific to Bucks here: viewtopic.php?f=12&t=36418