Add new 4x2 joists to the top of the old ones, inline, attaching with
glue and screws. This is legal, and will bring a dramatic increase in
strength to your floor/ceiling structure,

This is a well accepted method when the existing ceiling rafters are
already almost deep enough.

The aim of gluing and screwing the two parts together is to prevent the
composite beam from bending by one part sliding over the other. The glue
is much more important than the screws because the large bond area
distributes the shear forces more evenly. The main contribution of the
screws is to help make a good glue bond, and afterwards they are almost
incidental.

In loft conversions, this is usually supplemented by glueing and
screwing the chipboard floor down, which adds yet more stiffness to the
whole structure.



--
Ian White