There is some kind of an inflatable thing called "Plane Pal". It's basically an inflatable pillow that plugs into the space between your seat and the one in front of you (it does not look like it interferes with recline of the seat in front).
https://www.planepal.co.uk/
I have never used it so I cannot comment on how well it works etc.
BA is
not listed as one of the airlines that allow use of this item (
https://www.planepal.co.uk/pages/airlines), so you are well advised to seek a prior approval from BA.
If you can get BA's approval, this may be a potential solution to consider (the child will need his/her own seat, and must be window or middle, not the aisle)?