The structural properties of three- and four-site water models are improved by extending the ForceBalance parametrization code to include a new methodology allowing for the targeting of any radial distribution function (RDF) during the parametrization of a force field. The mean squared difference (MSD) between the experimental and simulated RDFs contributes to an objective function, allowing for the systematic optimization of force field parameters to reach closer overall agreement with experiment. RDF fitting is applied to develop modified versions of the TIP3P and TIP4P/2005 water models in which the Lennard-Jones potential is replaced by a Buckingham potential. The optimized TIP3P-Buckingham and TIP4P-Buckingham potentials feature 93 and 98% lower MSDs in the OO RDF compared to the TIP3P and TIP4P/2005 models respectively, with marked decreases in the height of the first peak. Additionally, these Buckingham models predict the entropy of water more accurately, reducing the error in the entropy of TIP3P from 11 to 3% and the error in the entropy of TIP4P/2005 from 11 to 2%. These new Buckingham models have improved predictive power for many nonfitted properties particularly in the case of TIP3P. Our work directly demonstrates how the Buckingham potential can improve the description of water's structural properties beyond the Lennard-Jones potential. Moreover, adding a Buckingham potential is a favorable alternative to adding interaction sites in terms of computational speed on modern GPU hardware.