Amphiphilic, monolayer-protected gold nanoparticles (NPs) have been shown to enter cells via a non-endocytic, non-disruptive pathway that could be valuable for biomedical applications. The same NPs were also found to insert into a series of model cell membranes as a precursor to cellular uptake, but the insertion mechanism remains unclear. Previous simulations have demonstrated that an amphiphilic NP can insert into a single leaflet of a planar lipid bilayer, but in this configuration all charged end groups are localized to one side of the bilayer and it is unknown if further insertion is thermodynamically favorable. Here, we use atomistic molecular dynamics simulations to show that an amphiphilic NP can reach the bilayer midplane non-disruptively if charged ligands iteratively ``flip'' across the bilayer. Ligand flipping is a favorable process that relaxes bilayer curvature, decreases the nonpolar solvent-accessible surface area of the NP monolayer, and increases attractive ligand-lipid electrostatic interactions. Analysis of end group hydration further indicates that iterative ligand flipping can occur on experimentally relevant timescales. Supported by these results, we present a complete energy landscape for the non-disruptive insertion of amphiphilic NPs into lipid bilayers. These findings will help guide the design of NPs to enhance bilayer insertion and non-endocytic cellular uptake, and also provide physical insight into a possible pathway for the translocation of charged biomacromolecules.