Would the solid material inside an airship displace air and be counted towards lift?
It would, but it is so little it is not noticeable
Duraluminium has a density of 2780 kg/m3, plain air has a density 1.3 kg/m3. So the buoyancy is in the order of 0.05%. The dry weight of the Hindenburg for example was 100 tons, meaning that the buoyancy of the solid material is about 50 kg.
This is so little that other effects on lift — i.e. changes in temperature, internal gas pressure, external atmospheric pressure, dynamic lift from relative wind speed, ballasting, spending fuel, and more — will completely drown out this effect.