They are really getting desperate with their claims. Their Notice to Keeper (NtK) is not PoFA compliant so no Keeper liability. No contract could be formed with prohibitive signage. No evidence of a contract being formed beyond the minimum consideration period, and I could go on, but I really can't be bothered.
Whilst they label this area as a “red route”, this has no statutory meaning on private land and is simply their own terminology, which is open to challenge. They can call it a "polka-dot route" as far as the law is concerned.
Your appeal is perfectly valid and, as any initial appeal is futile, don't spend any more effort on it. As this bottom-dwelling firm is an IPC member, don't pin much hope on the IAS. We'll give you a good IAS appeal, just because we love to wind them up by making them have to jump through the hoops.
Whilst this is most likely going to end up as a county court claim, it will never reach a hearing and will either be struck out or discontinued. You just have to follow the advice we give you here.