If it was me, I would fill out the appeal and send it within the time allocated (recorded delivery)
You need to complete that within the time period anyway. Take your time and fill out to your best.
From what you've shared, I do not think the appeal will take place, but with UKBA who knows.
Be prepared to make a full claim for any costs incurred. Keep a written record with proofs of those costs.
I'm expecting the caseworker will get the decision reversed pretty fast as it's very clearly a very poor and bad mistake.
You're not the first and will not be the last. It's a disgrace. I feel ashamed at how you've been treated