So sorry to hear that your DD hasn't been offered a Reception place at all - it's a horrid situation to be in.
We're in Lewisham, so not far from you. Four years ago DD1 was offered a place at a school that was about 2 miles away (our first choice, where she was at nursery was 650m from us) but, thanks to the positioning of trainlines, would have taken about an hour to walk to. The alternative was to catch two buses which wouldn't have been feasible come September as DD2 would be three months old and getting prams/buggies on buses for the school 'run' is a nightmare. Childcare once I returned to work (as a teacher) from maternity leave would have been impossible; no childminder within walking distance of our home would consider a drop-off etc at our allocated school.
So, my initial response was to cry lots but then to keep my fingers crossed that Bexley borough would come up trumps - this was before all the boroughs were fully in sync - and that DD1 would be offered a place at a school more than 5 miles away but opposite the secondary that DH teaches at. Meanwhile, I rejected the Lewisham school and said I'd defer entry until DD1 reached statutory school age (the term after her 5th birthday which, for her, was January 2010). This confused the admissions team as did most legalities associated with admissions or so it seemed so I told them to tick the 'home school' box and to put DD1 on the waiting list for every one of the 17 schools that was closer to us than the allocated one.
Bexley did come up trumps and DD1 was offered a place 5.something miles away which we accepted, whilst remaining on the waiting lists of 17 schools. Weekly phonecalls to the admissions team eventually resulted in the news that DD1 could be offered a place at our closest community school which, although it wasn't where we wanted her to be at all, it was local and do-able. At that point we accepted that place and contacted Bexley to turn down theirs.
Fast forward to September and DD1 starts Reception at our closest school. She is 10th on the waiting list for the school where I teach/she was at nursery. Weekly phonecalls continue.
By December she is 3rd on the waiting list for 'my school'. I learn that a child in Reception is moving to another very local school as the result of fraudulent behaviour by a parent whose child was due to start in January. I ring admissions with the name of the moving child and ask where that puts us on the waiting list. I even ask about the January-intake child who is currently being assessed for a statement of SEN and whose parents want to name 'my school' in the statement i.e. that child jumps the waiting list. Cue enormous cock up by the LA who tell me that they will be offering the place to DD1 as that child can be admitted as an 'excepted pupil'. I immediately e-mail and seek written response that DD1 will be offered place officially.
January 2010 arrives and I receive a phone call from 'my school' advising that the LA have over offered the vacancy - not only has it been offered to DD1 and the child with a statement but also now to the child with the fraudulent parent. 1 vacancy but 3 children to fill it so the LA want to withdraw DD1's place even though the fraudulent parent hasn't actually provided the evidence required for a priority place at 'my school'.
In the end, DD1 started back at 'my school', rejoining her nursery friends, rather rapidly - the second day of term instead of the original plan which was for her to do two days at the local school (to say goodbye properly and for coat pegs etc. to be set up at 'my school') - so that she had started before the January intake home visits happened/children started. Eventually the LA admitted DD1 as the child filling the vacancy, the child with the statement as an 'excepted pupil' and the child of the fraudulent parent as an 'excepted pupil' too. By the end of Reception the child with a statement had moved to a special school and the LGO overturned the ruling that fraudulent parent's child should lose a school place that had been obtained fraudulently so that child moved to the other very local school.
It was hellish and I sympathise with you trying to navigate the ins and outs of how to secure a place that's workable longterm and shorterm, for the whole family. What I would say is, this is London: there is a massive amount of movement both on waiting lists and within the schools themselves. Accept whatever apparently madcapped scheme that Bromley offer and contact them weekly for an update on where your DD is on the many waiting lists she is on. You will probably receive 6 depressing letters that state 'you applied for XX school but we could not offer you a place. The furthest place offered was 30cm and you live 92.5cm away as measured by our special triangulation system that isn't the same as Google maps or anything you can get your hands on'.
As Bromley measure proximity 'in a straight line from home to school' doublecheck that they are using 'as the crow flies' rather than 'by safest walking route' then borrow a trundle wheel* and measure the distance from your front door/boundary to the front entrance of Balgowan. Last year's furthest place was approx. 482m and, although that is not an indication of what this year's will be (one year it was 161m) it might make you feel better to know they haven't made up the numbers/to spot a mistake!
*I'll bring one home from school if you need it .
Good luck with it all.