We study the dynamics of flexible, semiflexible and self-avoiding polymer chains under the Kramers metastable potential. Due to thermal noise the polymers, initially in the metastable well, can cross the potential barrier, but these events are rare at low temperatures. To speed up the slow rate processes in computer simulations we employ the hyperdynamics method using the path-integral representation of the relevant Langevin dynamics . In this study, we extend the method for many-particle systems with internal degrees of freedom, such as the polymer chain. We study the regime where the well size is comparable to chain length. We find that the flexible, semi-flexible and self-avoiding chains exhibit qualitatively different behavior. For the flexible chain, the crossing rate decreases monotonically with the polymer length (L), while for the semi-flexbile chain the rate saturates at a level that depends on the chain stiffness. For the self-avoiding chain, on the other hand, the rate varies non-monotonically with L. For L less than Lm, the rate decreases as L increases, while for L > Lm, the rate increases approximately linearly with L. We attribute this behavior to the coil-to-stretch transition of the chain, which lowers the effective free energy barrier and enhances the crossing rate. This effect can be instrumental in efficient separation of biopolymers.  L. Y. Chen and N. J. M Horing, J. Chem. Phys. 126, 224103 (2007).