The only reason Young Adult exists is for the UK departure tax exemption. Over 12s pay adult fare but (and on UK departures in economy only) no APD. You will have no problem if you book one as an adult but you will very likely have an issue if you book them as a child, as once you enter DOB it will fail the next ticket audit check and could be cancelled without warning. It’s considered fraud so yes they can do this.

If ba.com isn’t playing ball you should either call to book, where fees will be waived, or book as an adult and call to change it later - in which case you’ll get a refund of any APD paid.

ba.com does normally work properly for YA bookings, so hopefully it’s only a temporary glitch.