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We study the strong-coupling (SC) interaction between two like-charged membranes of finite thickness embedded in a medium of higher dielectric constant. A generalized SC theory is applied along with extensive Monte Carlo simulations to study the image charge effects induced by multiple dielectric discontinuities in this system. These effects lead to strong counterion crowding in the central region of the intersurface space upon increasing the solvent-membrane dielectric mismatch and change the membrane interactions from attractive to repulsive at small separations. These features agree quantitatively with the SC theory at elevated couplings or dielectric mismatch where the correlation hole around counterions is larger than the thickness of the central counterion layer.