Yes think I’m done for and gonna have to pay up, wish I’d just paid the ticket bloke now but he sed I could dispute it
Unfortunately, you chose to pay a cheaper fare from Edinburgh Park, which was valid on specific trains only, so you have broken that rule.
You were offered the chance to pay the correct walk-up fare (which I make to be £16.10 as you had a ticket from Berwick*) and declined.
https://www.brfares.com/!fares?orig=EDP&dest=BWK&period=20230903.
Cross-country are therefore entitled to pursue this.
So therefore do I write back to them again disputing this and showing my tickets as proof
I think you have grounds for challenging the fare demanded, which is 2-3 times more than it should be.
The admin fee is on the high side, but I don't think you will get anywhere with that.
(I presume there was a previous letter to which you gave a reply which effectively admitted you had got on the XC train with an invalid ticket?)
You could therefore write back saying that you had a valid ticket from Berwick to Alnmouth (attach it, but no others) and therefore the fare owing is from Edinburgh Park to Berwick only, and request them to amend the settlement fee accordingly.
However, you need to make the letter apologetic as you were in the wrong.
As for the question of jurisdiction, I would only pursue that after taking legal advice.
BTW - I presume you missed the 1646 train at EDP? Is that correct?
(*for reference, the walk-up fare to Alnmouth was £25.80, only £4 more than you actually paid

)