Welcome to the forum.
From what you say, the railway will have no difficulty in getting a conviction: as you didn't have a Railcard, you didn't have a valid ticket, and as you knowingly bought an invalid ticket, you showed intent to travel without paying the right fare. So this post is written on the assumption that you are guilty of an offence under the Regulation of Railways Act.
What will be the consequences of being convicted? It will cost you quite a lot of money - the fine, court costs and compensation to the victim. Think some hundreds of pounds. You will, as you mention, also end up with a criminal record which will come up on DBS checks, so you won't be able to hide it from employers.
But it seems to me that this will be unlikely to be career finishing (always assuming that this is your only conviction). Many employers will see a conviction for fare dodging as evidence of one occasion of bad judgement. Repeat offences - or trying to cover up one conviction - might look like a pattern, at which point they may be more worried.
So even a conviction isn't a total disaster. But is there a chance of getting a settlement out of court? It varies by railway company, but there is a chance. The letter you have received sounds like either the first or second letter that Transport Investigation Ltd (TIL - they are contractors who do the initial investigation for a lot of railway companies) send. The first letter asks for your side of the story: the second essentially says they are handing the case back to the railway for prosecution.
It's the railway who decide whether to prosecute or not. So reply to this letter - and any following correspondence - briefly, politely and apologetically. Make it clear that you know you were in the wrong, that you have learnt your lesson and you won't do it again. Don't bother explaining that you were broke and that a conviction will have bad consequences for your job - you knew this when you fare-dodged so presumably decided it was a risk worth taking. But explain you are keen to settle out of court.
As I said at the top of this post, it looks to me as if you are guilty. So you are not in a strong negotiating position. But keep asking, and you may be offered a chance to settle. If you are offered a settlement, take it even if it is for a lot of money: in practice you won't be able to negotiate anything cheaper.
