From what I can see, what you signed for (on the plans shown) was a slope of 450mm from the path to the side of your house and a further 350mm to the boundary.
My advice is don't waste money on legal action when you can spend it on a decent landscaper to level up your garden with dwarf retaining walls and guarding for your daughter.
It is Persimmon and they don't care about anything. If you have additional issues then it may be worth going to court.
As for the NHBC claiming it is a "contractual issue" and nothing to do with them, I would point out that in most new build contracts it states that work will be done in a workmanlike manner in accordance to the approved plans and to meet all applicable Building Regulations and warranty standards.
Obviously, when a house has defects it is a contractual issue so, from what the NHBC said they would never pay for any remedial works under the warranty on any new home?
The NHBC inspector should have assessed the slope and made a judgement based on the warranty standards guidance and common sense.
For anyone reading this thread the message is clear, if you a Persimmon new home, don't expect them to be very helpful fixing defects!