"Not given notice to her current school as yet due to her not having anywhere to move to."
This will mean that you stand to lose a terms fees in lieu of notice (check your contract now) because you have to take up a state school place in a timely fashion, which usually means under 2 weeks.
If the LEA offer you the nearby school/s (which you hate) then if you turn it down, they have no obligation whatsoever to come up with another offer for you, nor can the FAP be activated on your DD's account.
The LEA have the responsibility to ensure that every applicant for a state school gets a place - not necessarily one of their preferences - but a place somewhere, so they have a role in co-ordinating admissions for all maintained schools (running the main admissions rounds, co-ordinating with VA schools and now academies as well, which are their own admissions authorities, handing the initial in-year application form, as you can name community, VA and academy schools all in one form)
If the desirable school is already 2 over numbers, then they need 3 current pupils to leave before they can admit from the waiting list. The only other way to get a place is to appeal for one.
(My guess is that the detailed section on bullying might be of more relevance if you were leaving one state school for another)