And am I allowed to appeal for a second time?
If the council could issue the penalty and then be judge, jury and executioner it would make a mockery of your right to appeal, which is enshrined in Article 6 of the European Convention on Human rights (https://www.legislation.gov.uk/ukpga/1998/42/schedule/1/part/I/chapter/5):
In the determination of his civil rights and obligations or of any criminal charge against him, everyone is entitled to a fair and public hearing within a reasonable time by an independent and impartial tribunal established by law.
This appeal will be quite straightforward as I said, i suspect that once the council are notified of the appeal they'll decide to give up. They invariably rely on the fact that members of the public don't know the law and basically they just hope that you'll give up. The full process is explained here:
https://www.youtube.com/watch?v=IR9CF7Z-wWw
This is the sort of case that should take me all of five minutes.