The policies seem clear to me -
Passengers aged between 12 and 16 years who are not travelling as an unaccompanied child require their own photographic identification.
I expect that the Customer Services rep, thinking the child didnt have any photo ID, suggested a debit card to ensure they could travel. Unfortunately the second rep felt that this wasnt appropriate and therefore the unaccompanied minor service now needs to be paid for.
Just get the child a citizen card and then you wont have to ask these questions for future BE domestic services.
It's not a service id pay them for!
It is a matter of personal choice. I wouldnt let my child travel on their own out of principle - a 13 year old would not neccesarily feel comfortable if the aircraft had a problem or was diverted to an airport which they had never been to before. At least with the service paid for, the staff have a process in place to deal with them.