I'd also say go for Burnham.
I'm in the sixth form at Beaconsfield high but was at Burnham Grammar from years 9-11, although I loved my time at Burnham Grammar it did not offer the subjects and extra curricular activities that I wanted for the sixth form.
If you move to Burnham your daughter could go to Burnham Grammar or Beaconsfield High but beware people from Burnham are being turned away more often for Beaconsfield High because they are so oversubscribed, so you could choose somewhere closer to Beaconsfield if you are set on the Bucks Grammar schools for example Farnham Common, where it is Burnham Grammar and Beaconsfield High catchment.
Could I ask whereabouts you live? We are in Iver Heath which is (ha ha) in catchment for Beconsfield High - in reality there is absolutely ZERO chance of getting in from here.