@Atlanta04 by aligning with our values, do you mean things which don't actually form part of the concrete, legal, oversubscription criteria of the schools you applied for - e.g. they don't allow pupuls to have a mobile phone, even on the journey to school - and your DD was not ranked highly enough (or anywhere close, if shes in the 90s on waiting lists) against those schools' actual oversubscription criteria to be offered a place? Or things which are part of the actual oversubscription criteria, but your DD still wasn't ranked highly enough to be offered a place (e.g. a faith school where your DD is of the same religion as the school, and you provided the required evidence, but although the cut-off was within the criterion she was in, she was ranked below on the 'oversubscription within criterion' grounds, or she was ranked in a criterion lower than the cut-off one)?
In your position, I would start with looking at schools which are on a single bus route from home. You already know that on March 2nd, the nearest school within your borough with a place available was the one which you turned down, but in he intervening weeks, places may well have become available at schools more convenient to get to (and, which - who knows? - may not be so antithetical to your personal beliefs), which you hadn't previously applied for.
Was the school you turned down the only one in the area which your DD would have been offered even if it hadn't been under PAN?
Providing DC with an education is a parental responsibility - you can do this by asking your home LA to find a school place, by submitting your list of preferred schools once your CAF - but they have already done that and you have declined the only offer that they was able to make under the Admissions Code, so it is now up to you to apply elsewhere and to appeal for as many schools as you wish which have not offered a place. But elective home education is just another valid choice, not a Gotcha! for an appeal.