In my experience, the best option (other than 2 business class seats, which would be expensive) would be to remain in economy and get a bassinet (infants have priority for bassinet seats)
But if the child is not going to fit (plus there's the hassle of buckling/unbuckling each time the seat belt sign comes on) and since you already have an extra seat, either take a airline-certified car seat on board, or request Qatar Airways for a child-restraint-device (I've used it a few times but have always had to remind the crew about it when boarding, and sometimes the crew didn't even know how to fit it to the seat). In both cases you'd have a multi-point harness rather than just the lap belt for the child

the car seat would definitely be more comfortable than the CRD though (more head support), especially if the child is already used to it.
one plus of the CRD is that the meal tray works with it, but with a 1-year old that won't matter.

either way, request the bulk head row as the floor space is great for crawling (and even sleeping, if you put a blanket on the carpet - even at 4+ our son has done that)
(you can't get an exit row when travelling with an infant/child)