We investigate the elliptical instability of a strongly asymmetric vortex pair in a stratified fluid, generated by the acceleration and deceleration of the rotation of a single flap. The dominant parameter is the Froude number, Fr=U/(NR), based on the maximum azimuthal velocity, U, and corresponding radius, R, of the strongest vortex, i.e. the principal vortex, and buoyancy frequency N. For Fr>1, both vortices are elliptically unstable while the instability is suppressed for Fr<1. In an asymmetric vortex pair, the principal vortex is less – and the secondary vortex more – elliptical than the vortices in an equivalent symmetric dipolar vortex. The far more unstable secondary vortex interacts with the principal vortex and increases the strain on the latter, thus increasing its ellipticity and its instability growth rate. The nonlinear interactions render the elliptical instability more relevant. An asymmetric dipole can be more unstable than an equivalent symmetric dipole. Further, the wavelength of the instability is shown to be a function of the Froude number for strong stratifications corresponding to small Froude numbers, whereas it remains constant in the limit of a homogenous fluid.