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We address the generation of initial conditions (ICs) for GRAMSES, a code for nonlinear general relativistic (GR) $N$-body cosmological simulations recently introduced in Ref. [1]. GRAMSES adopts a constant mean curvature slicing with a minimal distortion gauge, where the linear growth rate is scale-dependent, and the standard method for realising initial particle data is not straightforwardly applicable. A new method is introduced, in which the initial positions of particles are generated from the displacement field realised for a matter power spectrum as usual, but the velocity is calculated by finite-differencing the displacement fields around the initial redshift. In this way, all the information required for setting up the initial conditions is drawn from three consecutive input matter power spectra, and additional assumptions such as scale-independence of the linear growth factor and growth rate are not needed. We implement this method in a modified 2LPTic code, and demonstrate that in a Newtonian setting it can reproduce the velocity field given by the default 2LPTic code with subpercent accuracy. We also show that the matter and velocity power spectra of the initial particle data generated for GRAMSES simulations using this method agree very well with the linear-theory predictions in the particular gauge used by GRAMSES. Finally, we discuss corrections to the finite difference calculation of the velocity when radiation is present, as well as additional corrections implemented in GRAMSES to ensure consistency. This method can be applied in ICs generation for GR simulations in generic gauges, and simulations of cosmological models with scale-dependent linear growth rate.