I sadly don’t have any active BA bookings at the moment, but if you click towards the top in MMB it should allow you to select the old style. Then under each flight it should give an option for more details and the fare code will be in there.
Alternatively go to manage my booking on the Finnair website and ask for an email receipt. This is in a lot of detail and will include a column for “Class” which will be the fare code.