So let me get this right the FCA (which as I understand is the ombudsman) have instructed Barclays to reduce interest debt by £5k ?
TIX have confirmed the debt was repaid in full ?
Barclays won't refund the money and are saying go to the ombudsman ?
but you have been to the ombudsman already ?
So I think it would be fill in the forms for a small claim court with a statement and proof of all the above, the judge will more than likely award in Your favour, Barclays will have to pay costs and possibly come compensation if the judge thinks appropriate, if you lose, highly unlikely on the face of it you only have your own small claim court fees to pay (not much). I have been through small claims a few times.
In reality once Barclays get served with the case they will probably cough up immediately and it won't get to court, or they will ignore it and the judge will award in your favour, then comes the fun bit of forcing Barclay to pay with a court order, the world is your oyster then......
The only thing I can think of that may be a spanner in the works is if Barclays contest that £5k should go to someone else, another creditor or have of it to the IP for fees or similar.
Small claims court is a great experience and not harrowing it is just sat in judges room at a table and I have always found it a pleasant exciting experience and brilliant for serving a large dose of Karma !
https://www.gov.uk/make-court-claim-for-money/overview
https://www.citizensadvice.org.uk/law-a ... aims-track
You might want to draft the particulars of claim and send to Barclays legal department stating unless paid within 7 days these will be filed at court, they might pay it then.
Good luck.