Underdoped cuprate superconductors exhibit high pairing energies, but low critical temperatures due to strong phase fluctuations in the pairing order parameter. It has been demonstrated experimentally, and studied theoretically within a mean-field framework, that it is possible to overcome these fluctuations by coupling the superconductor to a metallic layer. Beyond the mean-field level, we show using Monte Carlo simulations, that fluctuations considerably renormalize the superfluid density. An effective weak coupling theory is constructed in order to explain this within an analytical framework.