In one were to have incorrectly purchased said ticket after drinking just a little bit, leading to the momentary lapse of judgement, would it be a good or bad idea to include this in the return letter?
Unless it has changed dramatically since I was at university, the general position is that voluntary intoxication is no defence, not least because of the public policy interest in not giving defendants a trivial escape from the charge(s) they face.
In a relatively narrow range of situations, voluntary intoxication can negate
mens rea - if you were so drunk that you could not form the necessary intent for a crime of specific intent (one where the requred
mens rea is intention, such as the RoRA offence you appear to be accused of) that can lead to conviction of a lesser charge which, in this case, would be the byelaw offence (a strict liability offence, meaning it has no
mens rea element).
To my mind, your statement when stopped that you stated a false origin station to save money, that you make a mistake and that you regret your action appears to be a clear admission of an intentional act designed to withhold money rightly owed to the rail company. Any poor judgment behind that intention arising from voluntarily drunk alcohol is your fault and your responsibility. As I understand the law, you would have to be so drunk that you were incapable of making a deliberate decision to "short fare" for intoxication to come into play - and if you succeeded in this argument you would still likely be pursued for the lesser byelaw offence. If you were that intoxicated it would likely have been noted by the official who stopped you. Also, as you suggest, being that intoxicated would potentially have put you in the position of facing action for the byelaw 4(1) offence of being unfit to enter or remain on the railway owing to intoxication.
If you want to try to settle this matter out of court, a full and frank admission of guilt might fare better than trying to argue against your admissions at the scene on the basis of intoxication. The rail company or, if it gets that far, the prosecutor does not have to exercise its discretion in offering an out of court settlement, but is perhaps more likely to do so if it feels you show insight into your wrongdoing, recognise the problems the railway faces from fare evasion and are a low risk of repeat offending. Indeed, arguing against your admissions whilst trying to settle out of court gives mixed messages - on one hand you are arguing that, at most, you are guilty of a lesser offence whilst on the other you are fully admitting guilt and offering to make restitution without the matter proceeding to prosecution.
If you are ultimately convicted in relation to this matter and are worried what effect this will have on your possible future, I suggest you look at some of the decided cases before the Medical Practitioner's Tribunal Service. Those doctors who admit their wrongdoing, show insight into the causes of their wrongdoing and take steps to address those causes tend to fare much better than those who argue against the weight of evidence against them, doggedly refuse to admit to their wrongdoing and deny their wrongdoing caused actual or potential harm.
A former GP of mine got himself involved in an inappropriate relationship with a patient - something for which he would most likely have been struck off. By the time the case came before the GMC (who heard disciplinary cases back then) he had ended the relationship, resigned from his role as a partner in the surgery, made a full and frank admission to the GMC of what had happened, sought professional help for the underlying problems he was facing in his life, and took full responsibility for the harm his wrongdoing had done. He was subject to a sanction short of striking off, but I think this was a relatively short suspension from the medical register.
Conversely, a single, isolated dishonest act from a doctor who denies they did anything wrong and shows no insight into the harm their wrongdoing has caused can lead to them being struck off, as dishonesty accompanied by a lack of insight is typically felt to be fundamentally incompatible with the trust placed in a registered medical professional.
It might be that tougher standards apply to those seeking to enter the medical profession than to those already in it; I simply do not know. If you are ultimately convicted you would be best to enquire with the General Medical Council and/or any university you intend to apply to if you do wish to pursue medicine. Bearing in mind the negative consequences of conviction, it might be worth engaging a solicitor if prosecution is unavoidable - though, as you recognise, you have already made various admissions that seem likely to limit the scope for defence or mitigation. My understanding is that an enhanced DBS check is part of the process of applying for medical school.
Humans are fallible creatures and sometimes we make bad decisions. A clean record is ideal, but just because you have something in your past that is disclosable doesn't mean there is no chance of entering a particular profession, especially if it was an isolated incident, you disclose the matter at the earliest opportunity and make it clear that you learned your lesson. As has been said by others, not disclosing a relevant conviction that subsequently comes to light is likely to be gross misconduct justifying dismissal even if disclosing that conviction when asked would not have precluded your appointment. In this scenario, it is the dishonesty by failing to disclose a disclosable conviction when asked that is the gross misconduct.
Note: I am writing based on knowledge gained from studying law at university, but am not a lawyer and have no specialist knowledge of railway offences. Others responding to you in this thread are better placed to address your situation than I am. Nevertheless, I hope this is helpful and constructive.