We investigate in detail the sensitivity of the fourth-order charm fluctuation, calculated on the lattice, to the input bare charm quark mass. We approach its continuum limit by employing four different lines of constant physics. We quantify the cutoff effects arising due to bare charm quark mass for both coarser and finer lattices. Finally, we show that the ratios of generalized charm susceptibilities, calculated on coarser lattices, and the continuum-extrapolated fourth-order charm fluctuation are sufficient to obtain the continuum limit of all physical observables relevant to open charm physics.