>>> The website they referred to is infact a third party website and got nothing to do with Barclays. So they are asking me to apply through a third party loss and trace account/money
Assuming that it was
http://www.mylostaccount.org.uk it's an official site supported by the Building Societies Association, National Savings and Investments and UK Finance (which is the banking trade association). So there would be absolutely no risk involved in using it.
https://www.barclays.co.uk/current-accounts/dormant-lost-accounts
However, as you've said, you know the account details, so you shouldn't need to go through that service. Last year Santander marked one of my accounts as dormant and were about to do the same with another. I managed to get everything sorted out fairly easily though, simply by visiting a branch and getting so sit down with one of their managerial staff. That was down during one of the 'lockdown' periods, so I can't see why you can't do the same.
Contact Barlclays for a face-to-face (or should that be 'mask-to-mask'?) appointment!