PP, on the surface, is a bit of a grey area for a number of reasons, but for your circumstances, I would bring the schools attention to the following points:
The school should receive pp funding, if they didn't apply for it and you had given them all the relevant information then they are at fault, and your DD should not be disadvantaged by their failure to apply for the pp. I still raise an eyebrow at this!
I don't think the pp is statutorily ring fenced, but the accountability element IS answerable by the school, and in particular to OFSTED. Schools, as of Sept 16, have to state on their website how the pp is spent, and how the barriers to learning (social, emotional, educational) have been overcome to achieve the required learning outcome. Any sensible school will draw up the plan with the parent, as success needs to be measured (by parents and the school) and the desired outcomes verified.
Any school that chooses to ignore this, and uses the money for the good of all pupils when there is clearly a need for the child who has bought the funding to the school, to have additional support, is frankly, heading for trouble as pp in schools can have a review forced on the individual school by the LA, OFSTED, Dept of Ed. et al if it is felt the pp is not being used effectively. It is this accountability element that effectively ring fences the money as funding for your DD.
It is the accountability element that you must drive home, they will need to show they have assessed her needs, drawn up a plan to address the barriers to learning and measured the success of the spending.
A lot of schools have already put details of their pp spends on their website as good practice, although they will have to do as such this year. I am sure you can find examples if you google.
For all the reasons above, I think your school stand a good chance of having the funds backdated if they push hard enough as pp is pushed so hard at schools and I can't see the LA disadvantaging, what they see to be an already disadvantaged child. The school might get their knuckles rapped for failure to apply, but that is not your issue.
I hope that helps. If you need any further advice on the pp I am happy to point you in the right direction if I can. Please do not let the school fob you off!