You have asked me to give you an opinion, though there is not much that I can add to what has already been said.
I do agree that the C.A.B. have an unfortunate tendency to misunderstand railway offences, and they can give completely wrong advice (based either on the wrong assumption that it will be a Police matter, or the wrong assumption that it is a civil claim). But in other cases, they have had success in negotiating a settlement with a train operator on behalf of a passenger.
I also agree that Passenger Focus have a tendency to refer passengers back to the Train Company to attempt to resolve any disputes themselves, but Passenger Focus will sometimes intervene on behalf of a passenger.
You should certainly attempt to reach an agreement with the Company to settle again. But if that fails, then you will have some options available to you:-
1. Admit guilt to both offences (i.e. to entering a train without a valid ticket and giving a false n & a). I expect that you don't want to choose this option!
2. Instruct a solicitor to act for you, a local law firm specialising in small-scale Criminal Defence work will be able to help. This will cost an amount in 3 digits - possibly more if you reach options 3.a or 4.b (below).
3. Admit to the first offence (entering a train without a valid ticket) but not to the second offence (giving a false N&A). This will require some evidence to show the information you had shown the inspector. You will be taking a gamble that the Company will accept this and avoid the time and expense of a second hearing with witnesses to dispute the evidence in open court.
You will be making this admission and refusal in writing.
3.a. If the gamble suceeds, then you will attend the Court just to confirm your name and your plea - there will be no difficult questionning. The Sentence will be a modest fine plus costs, which will not appear on a Criminal record check.
3.b. If the gamble fails, the Court will schedule another hearing at a later date - that will be more difficult but we can go into details about how to manage that nearer the time. The costs will be higher.
4. Deny both charges against you. This is most likely to lead to a second hearing, by which time the costs will be higher. At that hearing, the evidence against you is read out (perhaps by one of the Inspectors who spoke with you and took notes of your explanations and the difficulties with confirming your address) and where you or your representative can cross-examine that witness to expose any fault with their evidence.
4.a. I would not recommend this choice if there is a probability that the evidence of the Inspector will give the impression that you had given a false name and address. It is generally assumed that people who are unable to give the correct address immediately are attempting to mislead.
4.b. If the explanation of the difficulty with the address can be shown to be a simple error on the Inspector's part, then it will be worth pursuing. You haven't explained to my satisfaction that this is what happened - only you will know that.
If you choose the options 3.b or 4, then you will either have to represent yourself in open Court or use the sevices of a solicitor.
If you represent yourself, then you will be standing next to a professional Prosecutor who knows that the way to succeed with a Prosecution is to be sure of the Law, to be sure that the Evidence is consistent with the Offence that is written in Law, and who is familiar with the procedures in the Courts. You may win sympathy from the Magistrates but that sympathy doesn't affect how they must determine the outcome.
If you use the services of a local Criminal Defence solicitor, then the prospects of a defence based on strict legal and evidential grounds are going to be better.
One final suggestion to consider: If you are going to attempt to reach a settlement with the Company again, you could give them an indication of how you are likely to respond if they refuse. If you make it clear that you are willing to let them dispose of the matter by the payment of an 'administrative settlement', and that if they refuse, then you will choose option 3, then you will have given them a choice - a choice where they face the time and expense of providing evidence in a second Court hearing to argue about your address, or accepting a settlement. This negotiated gamble is something that even the CAB should be able to do for you!
I hope this helps for now.