There's a couple of different questions here:
a) what's a good backpack that is within BA's dimensions for a "cabin bag" (56 x 45 x 25cm) but will in practice fit under the seat in front of you?
b) is there a "backpack" that is within BA's dimensions for a "personal item" (40 x 30 x 15cm)?

To my knowledge, there's nothing that's a "backpack" and a serious travel item that's an answer to b). The only "backpacks" that I know of that are 40 x 30 x 15cm are designed for smaller children or are more fashion accessories than serious travel items. I think the best choice here is a bag that isn't a "backpack" but instead has a shoulder strap, such as the amazon item cws mentioned.

There are many possibilities that are an answer to question a). Personally, I use the Osprey Talon 22:
https://www.rei.com/product/177573/o...CERAMIC%20BLUE
(I'm linking to the REI page and not the Osprey page because the dimensions are incorrect on the Osprey page). When buying something like this, I always buy from a brick and mortar shop with a wide selection of different packs and take along a tape measure (so I can check for both comfort and size). Also, for this purpose, I avoid a bag with a rigid frame, so I can squash it a bit if necessary. Usually, there's enough space in an overhead bin for my Osprey Talon but it will fit under the seat in front of me if required.