We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N . Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order N 2 ), the system synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. In a previous paper, we showed that the transition from strong to weak coupling involves a sequence of symmetry-breaking bifurcations of the system's stationary configurations, and analysed in particular the behaviour for coupling intensities slightly below the synchronisation threshold, for arbitrary N . Here we describe the behaviour for any positive coupling intensity γ of order N 2 , provided the particle number N is sufficiently large (as a function of γ/N 2 ). In particular, we determine the transition time between synchronised states, as well as the shape of the "critical droplet" to leading order in 1/N . Our techniques involve the control of the exact number of periodic orbits of a near-integrable twist map, allowing us to give a detailed description of the system's potential landscape, in which the metastable behaviour is encoded.