I would ADVISE EXTREME caution against taking Y on a TATL trip on AC's 777 product with slim line seats.
If the front person lower his/her pitch to the maximum, and your daughter put out the backrest It would be EXTREMELY uncomfortable , esp during meal time.
I have seen passengers taking out the backrest and put it on the tray table after a while, but if your daughter is tall, she will not be able to rest her head on the seat when the person in front has reclined to the maximum.
I have zero back problem and still experience back pain one - two days after TPAC.
It comes down to: Is she tall?If she is chances are she will be more uncomfortable with it.
Please pay for PE for her sake, I have relatives with back issue and its not worth saving just a few hundred with the possibility of entire trip ruined.