I have had a read of your story and hopefully I'll be able to give you a bit of advice. Essentially you have two options available to you if you want to go the litigation route. Your first option would be to bring a claim in tort based upon negligence. You could do this because a bank owes a fiduciary duty to its customer(s) to use reasonable skill and care when dealing with them. They are also subject to the duty of care that everybody else is subject to. So in order to prove a duty of care is easy and it is also easy to show that they failed to use reasonable skill and care. However in order to successfully prove a negligence claim you need to be able demonstrate causation and forseeability, this could be more complicated even though it is clear they have been negligent.
The second much easier option is to make a claim in contract. You would simply have to prove they are in breach of contract, this is easy. Implied into every contract, regardless of what is in the small print of your terms and conditions, is a term that the supplier of a service will use reasonable skill and care when supplying the service. This is found in s.13 of the Supply of Goods and Services Act 1982. In order to prove this you need to prove the existence of a contract: this doesn't need to be signed. You will need to check your terms and conditions for any other relevant terms as well as the implied statutory term. You will also have to prove a breach of contract; in your case they have taken money from your account without your consent and failed to use "reasonable skill and care". You will also have to show the court that it has caused you a loss, that will be easy just the court your bank statement detailing how much money they took.
If I were you I would definitely consider the second option as it is straight forward and very easy to prove in court before a district judge in a small claims court. Before starting proceedings though I would write a letter to your bank rejecting the initial offer of £150 as being unreasonable in the circumstances and state what amount you would prepared to accept and also specify a time limit when you need a reply back otherwise you will commence proceedings in the county court.
I’m new to this site as someone at work told me about it, I work as a solicitor in a City practice and no doubt you will read a lot of people on this site talking about the Consumer Credit Act and the like; this just complicates matters and makes things take longer. If you stick to basic legal principles of contract then it is usually very straightforward.