Is there a positive displacement compressor that can rival axial compressors in terms of air mass flow to weight ratio?
I'm going to say No.
Just because the simplicity of an axial compressor will always beat the complex piston system, unless your prime mover is somehow linear, I guess (A giant on a footplate?). So will centrifugal or screw compressors. Which is why we only see piston designs for low output, high pressure applications.