This paper entails the study of the pulsed-jet propulsion inspired by cephalopods in the frame of underwater bioinspired robotics. This propulsion routine involves a sequence of consecutive cycles of inflation and collapse of an elastic bladder, which, in the robotics artefact developed by the authors, is enabled by a cable-driven actuation of a deformable shell composed of rubber-like materials. In the present work an all-comprehensive formulation is derived by resorting to a coupled approach that comprises of a model of the structural dynamics of the cephalopod-like elastic bladder and a model of the pulsed-jet thrust production. The bladder, or mantle, is modelled by means of geometrically exact, axisymmetric, nonlinear shell theory, which yields an accurate estimation of the forces involved in driving the deformation of the structure in water. By coupling these results with those from a standard thrust model, the behaviour of the vehicle propelling itself in water is derived. The constitutive laws of the shell are also exploited as control laws with the scope of replicating the muscle activation routine observed in cephalopods. The model is employed to test various shapes, material properties and actuation routines of the mantle. The results are compared in terms of speed performance in order to identify suitable design guidelines. Altogether, the model is tested in more than 50 configurations, eventually providing useful insight for the development of more advanced vehicles and bringing evidence of its reliability in studying the dynamics of both man-made cephalopod-inspired robots and live specimens.