Will a 0.400mm thick PCB be too flexible?
I've done a 1 mm thick board with dimensions about 3 x 15 cm. This design was noticeably more flexible than a similar 1.6 mm board, and flexible enough that I worried about the possibility of board flex causing damage to mounted components or the solder joints holding them to the board.

Given that the flexibility of a board depends on the cube of the board thickness (because it depends on the area moment of inertia of the cross-section of the board), and your proposed stackup is less than half the thickness of mine, I expect you will need to take special care to mount this board at multiple points, or provide some other stiffening mechanism, to avoid damaging the board or the parts on it.