A novel method for generating GigaGauss solenoidal fields in a laser-plasma bubble, using screw-shaped laser pulses, has been recently presented. Such magnetic fields enable fast synchrotron radiation cooling of the beam emittance of laser-plasma accelerated leptons. This recent finding opens a novel approach for design of laser-plasma FELs or colliders, where the acceleration stages are interleaved with laser-plasma emittance cooling stages. In this concept paper, we present an outline of what a staged plasma-acceleration FEL could look like, and discuss further studies needed to investigate the feasibility of the concept in detail.