The discovery of high-$T_c$ superconductivity in pressurized bilayer nickelate La${}_3$Ni${}_2$O${}_7$ presents a new arena for exploring unconventional pairing mechanisms. A pivotal yet unresolved issue is the specific role of the $3d_{z^2}$ orbital of Ni. While its inter-layer super-exchange antiferromagnetic coupling is widely considered crucial for superconductivity, the role of its itinerancy remains undetermined. Early studies showed that the superconductivity is accompanied by the emergence of a small Fermi pocket of the $3d_{z^2}$ orbitals. However, recent experiments show controversial results on the role of the $3d_{z^2}$ Fermi pocket on superconductivity. Motivated by these experimental results, we investigate an effective bilayer two-orbital model for La${}_3$Ni${}_2$O${}_7$ using density-matrix renormalization group (DMRG) on a minimal one-dimensional geometry. By systematically varying the $3d_{z^2}$ orbital filling from $1/12$ doping to half-filling, we observe a pronounced suppression of superconducting correlations near half-filling. Our results demonstrate the itinerancy of $3d_{z^2}$ orbital is favorable for the pairing in the bilayer two-orbital model for La${}_3$Ni${}_2$O${}_7$. Moreover, we observe that the pairing correlation is enhanced in regions where charge fluctuations are large, suggesting a competition between charge order and superconductivity in the model.