Did you write and say you wished to be removed from the waiting list for burnham after accepting Becky high?
The problem, if you did not, is that the allocation process will have given someone else the opportunity for the Becky high place, after your first option of Burnham came up. Mind you Becky high had added a whole extra class, so there should have been plenty of spare places.
Saying that a week is not a long time to send a letter to remove your child from a waiting list, also the normal process is to allocate in rounds, not a week after round two.
I sounds like they may have made a bit of an administrative error? I would make sure there is a letter in the system, removing your daughter from the waiting list and saying you have accepted Becky high. Then stand firm, saying they did not give you time to send the letter before the unexpected follow up letter, which was issued prior to round three allocations.
Just a final thought, maybe consider why you put Burnham in the first place, if your life will be much easier without the extra 5 mile treck into beaconsfield, or if there are other strong reasons, you could perhaps work on your daughters voice of reason. After all she has pulled the woman's prerogative of changing her mind...