Yes, BD definitely makes sense for the SQ segments. Do double-check that the booking class is eligible for BD miles, though (see here for table of eligible booking classes). As to the QF segments, the problem with BA is that you will only get 25% mileage on discount economy. It is worth looking around at other OW programmes, AA in particular, to find out whether it would make sense for him. In any event, flights on BA itself in discount economy will earn miser miles, whichever FFP you credit it to.