Please do not bump your thread. This is a busy forum where volunteers provide advice for free. Sometimes you will have to wait for advice - the urgency you now face is due to us having to wait 6 days to see the Notice to Keeper, although I do appreciate the unfortunate circumstances that led to this, and I'm sorry for your loss.
There are a couple of arguments you can advance:
Forbidding Signage:
The signage does not make an offer to non-permit holders to park. Instead, it prohibits those without a permit from parking. Without this, there can be no contract formed, and thus no money owed. Neither ParkingEye nor POPLA will be likely to uphold an appeal on this basis, but it's an argument that could be advanced in court.
PoFA Compliance:
There is an argument that ParkingEye's notice does not quite comply with the requirements of Schedule 4 of the Protection of Freedoms Act, insofar as it does not invite the keeper to pay, as required by 9(2)(e) of the Act. Again, neither ParkingEye nor POPLA will be likely to uphold an appeal on this basis. To my knowledge it's not an argument that has yet been tried in court, as a good portion of the claims we see issued are discontinued for one reason or another before they reach a hearing.
Conflicting Signage:
You have shown us that the car park bears signage both for ParkingEye and FlashPark. You obviously cannot enter simultaneous contracts with two different firms for the same parking 'service', but you could seek to argue that the presence of conflicting signage makes the charge unenforceable, as it is not clear to a motorist which company they are entering into a contract with. It would be useful to get some additional photos of the signage at the site showing its location etc. to reinforce this point.