my experience is that the first appeal to the employer who sacked you originally is very rarely won(none in my working life) but are very often settled just prior to court, I suggest this will be how this one will go
Agree ... legal and lose is uber risky, even a win may do nothing to enhance reputation. Appeal actually becomes a sort of compromise negotiation, particularly agreement to what is said (or more importantly what not said) to the outside world. I've seen stuff written into Compromise agreements that Hans Christian Andersen couldn't have made up