The energy needed to create a vortex core is the basic ingredient to address the physics of thermal vortex fluctuations in underdoped cuprates. Here, we theoretically investigate its role in the occurrence of the Beresinskii-Kosterlitz-Thouless transition in a bilayer film with inhomogeneity. From the comparison with recent measurements of the penetration depth in two-unit-cell thin films, we can extract the value of the vortex-core energy μ and show that μ scales linearly with Tc at low doping.