Welcome to the forum Rebecca. From the pic it looks like a mk1 (1999-2006) so you are in the correct section if so. The answer is long and a bit uncertain, so bare with me,lol
AFAIK the
problem is in the cluster itself. I had the same
problem with mine, (apart from i didn't get the beeping) but mine got worse and worse until it got to the point where it was also strobing various warning lights when driving, and it was also starting to affect the speedo, rev counter and even the immobiliser in the end.
After some troubleshooting I found that it seemed to be temperature related as it was fine in the summer, but on a cold or damp morning i would get the issue you describe.
To test this out i started removing the cluster and taking it inside the house with me at night. When it was kept warm, then the issue never happened when it was reconnected, so i was almost certain this was the case.
On the upside, the lights (LED's) and buzzers are very low current, so unless it's sat for days without being driven then it shouldn't cause your battery to go flat.
The clusters themselves are over £400 from SEAT, (plus you will need to have it coded to the car by an auto-locksmith/SEAT) but i got lucky and managed to get a brand new one that someone had had in stock for years unopened. Since fitting it i haven't had any further issues, so this further confirmed my earlier theory.
If you don't use the car everyday, then there are companies out there that you can send the cluster away to, and they will repair it for you. This was going to be my next option if it hadn't found a new one.
http://www.clusterrepairsuk.co.uk/repairlist
https://www.autotronics.co.uk/
Like i say, this is all just from my experience with it, so although i'm 99.9% certain that everything i said above is correct, I obviously can't take responsibility if yours turns out to be something else after you spend money on sending your cluster off etc etc. Your other option is to take it to an 'auto-electrician' and get them to have a look, but i'm pretty sure they will come to the same conclusion. I hope that is some
help for you.