Sadly I would say give up and find a good Seat/VAG specialist who will know about the issue, and that the part is about £250+labour+VAT, which will be half what Seat will want.
I have posted here before about my two experiences, but not sure if I updated the last.
The first was my nearly new 07 Cupra that saw the fault about 65k in 2011/12. After alot of arguing with the dealer (different from where I had bought it as that one had gone bust!) I raised it with Seat UK who were quite helpful and I can't recall what I paid but I think it was £200 at most. I believe there was a tech bulletin floating around dealers that had the information about it all, same as for the soggy under-bonnet sound-proofing which they denied was an issue too (which I also got done FOC).
For various reasons I sold it soon after that and recently got a 57 FR that had only been about 6k a year, and last year got to over 60k and guess what, same again. Tried a polite enquiry to Seat UK, they said to get dealer diags (I had already had 3rd party diags FOC!) to send to them. Dealer wanted £90, I asked for a quote for what I believed the issue was BEFORE they did the diag on the basis that if it was that then they would knock the £90 off if I paid - if I couldn't get goodwill from Seat UK for some/all. They quoted "about £400".
The dealer did the diagnostic, said it was "an ABS valve issue" but couldn't confirm exactly what as the mechanic was not in by the time I collected it... I called Seat UK, they got the information from the dealer and said it wasnt a known issue and didn't want anything to do with it. The "entertaining" bit was when I asked for a copy of the diagnostic report. They refused and told me to go back to the dealer. The dealer said they only have paper copies and had thrown it away. After a "discussion" with the service manager he amazingly asked for £90 to give me the report (not realising that I had not actually been charged for the first as it was pending the response from Seat UK). I said I had already paid for one and not had it, he said he couldnt just allocate time and not have it paid for... Seat UK incidentally denied there ever was a fault/issue and couldn't find any record of me talking to them about my first car!
So I asked for a proper quote to be emailed to me (to try and prove it was the part that has a known fault!), which was a screenshot of their billing system and the line "ABS valve replacement" and now at £800, but a £1200 option for "whole block" to be replaced...
By this point I had it fixed at a local specialist who knew all about them and had done many.
I persisted for a proper itemised quote from the dealer, eventually making a complaint to their principal who agreed it was not acceptable service... The proper quote was slightly higher again, so I complained for the hell of it and asked what the difference was between my (20+years at VW) specialist doing it and them, and to be fair he didn't try and make too much up to counter that.
About a month later I heard they went bust.
So... Still with me?
In summary, from what I have heard here and experienced, the
problem only shows after about 5 years, which I guess is ~12k miles a year (average for alot of people I guess these days) going on my first car, and even though my second was getting on 9 years, at 60k on my second.
77k you have done well in my opinion, just don't let a dealer rip you off. They will claim the diags take quite a long time - it's not worth the money. I have sat there with someone and it took longer for the laptop to boot than it did for it to read all the data!
BTW - just to check, did you notice that your traction control was the
problem, but your ABS still worked fine?