but the roof is quite shallow, and at its highest point is only about
5'11''.

About your only chance is to lower the ceilings below if there is any space.
And use steel joists, instead of wood! My joists can go as low at 76mm using
steel, as opposed to 220mm using wood.

You're going to lose around 30mm off the roof line as well, unless you have
4"+ rafters.

Christian,