We employ the macroscopic fluctuation theory to study fluctuations of integrated current in one-dimensional lattice gases with a steplike initial density profile. We analytically determine the variance of the current fluctuations for a class of diffusive processes with a density-independent diffusion coefficient. Our calculations rely on a perturbation theory around the noiseless hydrodynamic solution. We consider both quenched and annealed types of averaging (the initial condition is allowed to fluctuate in the latter situation). The general results for the variance are specialized to a few interesting models including the symmetric exclusion process and the Kipnis-Marchioro-Presutti model. We also probe large deviations of the current for the symmetric exclusion process. This is done by numerically solving the governing equations of the macroscopic fluctuation theory using an efficient iteration algorithm.