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In conventional gravitational physics, the so-called ‘bootstrap procedure’ can be used to extrapolate from a linear model of a rank-2 tensor to a full non-linear theory of gravity (i.e. general relativity) via a coupling to the stress-energy of the model. In this work, I extend this procedure to a ‘Lorentz-violating’ gravitational model, in which the linear tensor field and the matter fields ‘see’ different metrics due to a coupling between the tensor field and a background vector field. The resulting model can be thought of as a generalized Proca theory with a non-minimal coupling to conventional matter. It has a similar linearized limit to the better-known ‘bumblebee model’, but differs at higher orders in perturbation theory. Its effects are unobservable in FRW spacetimes, but are expected to be important in anisotropic cosmological spacetimes. [ABSTRACT FROM AUTHOR]