In my experience you've almost zero chance of winning an appeal pre-court with the parking company, they'll just ignore everything you say and continue with their threats.
No idea about what happens if it gets as far as court, or how likely it is that it will go that far.
In these circumstances, your best bet is usually to ask the landowner to overturn the invoice, they can do this, but sadly they're not co-operating, which is rather short sighted of them, in my experience they are happy to do this, especially if an error has been made and they don't want to upset a customer.
I don't normally review restaurants or use the power of social media to threaten business owners, but in these circumstances, I would politely ask the restaurant owner/manager to ask the parking company to cancel the ticket that was issued due to their error and if they don't, I will mention in my review of the restaurant that people should use the parking facilities with caution due to the risk of getting a parking ticket.
Because if you try and sort it with the parking company, you're facing months, if it not years of hassle that will leave you wishing you'd just paid the ticket.