Let's start with something that might be a side issue, but might be enough to make this demand for £2,198.72 go away.
If I have understood you correctly, this is in an email. Are you certain that it is from GWR? On the basis of cases we see here, when railways suggest a settlement they do it by a letter sent through the post. It's possible that things have changed so this email might be genuine, but please check the email address that this has been sent from. If this is someone either winding you up or trying to defraud you, then the way you need to deal with it will be rather different from what to do if this really is from GWR.
But for the rest of this posting, I'll assume that the email from GWR is genuine. And let's start by looking at what usually happens.
For some reason or another, GWR think that you dodged a train fare. That means that they are allowed to take you to court and prosecute you for not having paid the fare - and if they can persuade the court that you didn't pay the fare, then you will be found guilty and have to pay a fine, plus compensation (the fare in dispute) plus court costs plus (maybe) GWR's prosecution costs. But this is quite a lot of hassle for GWR, as well as for you, so what happens quite a lot of the time is that GWR will offer an 'out of court settlement'. That's what the request for £106 was. That was probably made up of something like £6 for the train fare, and £100 for the costs that GWR had incurred.
You haven't yet told us quite what happened when you deemed the £106 request to be unfair but it sounds as if GWR have now had another look at the information they hold and think that a higher figure might get you to talk to them. From what you've put on this discussion, I don't think that's worked properly: it's just made you rather cross.
So what happens next? If you don't manage to come to an agreement with GWR (and more about that later) then they will probably take you to court. By the sounds of things, you disagree with GWR that you failed to pay a fare, so at court you can challenge them to prove that you did what they claim you did. So if you haven't avoided fares on the dates that GWR say that you have, then you should be OK.
But I think there is still a chance to talk to GWR and come to an agreement. Before doing that, try and understand things from their point of view: in the original incident, did you pay the train fare or not? If you didn't, then it is quite likely that GWR have a point: the way the law is written, a passenger is meant to put in a lot of effort to pay the right fare. Broadly, if there was some reasonable way that you could have bought a ticket before the start of your journey (most obviously an open ticket office, but also maybe a ticket machine) then you should have done. And I think that if GWR gave you a chance to buy your ticket before you left the station at the end of your journey, then you should have taken that chance. If you had a chance and you missed it, then you may - in the eyes of the law - be in the wrong.
If you did have the right ticket, then you can be robust in your discussion with GWR: ask them to explain why they think you didn't have the right ticket, and (since you did have the right ticket) make it clear that you are prepared to argue the point in court and see GWR lose. But if you didn't have the right ticket then it might be that the £106 is the least you can get away with. Talk again to GWR: do so in writing (either a letter or an email) - and make sure that you are particularly polite and apologetic. I think that this might well work, in that I think that the point of the £2,198.72 demand is to frighten you and get you to come back and talk to them again.
At this stage, remember that until you come to a settlement with GWR (whether that's because they accept that they're wrong and walk away, or because you agree to pay them a settled amount) they have the right to take you to court. So you're not in a strong position. But I think it is still possible that you can sort this out without having to pay thousands of pounds.