The physical properties of solid-state materials are closely tied to their crystal structure, yet our understanding of how competing structural arrangements energetically compare is limited. In this work, we explore how small differences in composition affect the structure in the La(AuxGe1-x)2 series of compounds, which comprises four unique structure types between LaGe2 and LaAu2. This family includes the previously unknown AlB2-type compound with stoichiometry La(Au0.375Ge0.625)2 as well as La(Au0.25Ge0.75)2, an intergrowth of the AlB2 and ThSi2 structure types. We then study the chemical forces driving the structure changes and use phonon band structure calculations and DFT-Chemical Pressure to evaluate atomic-size effects. These calculations show that the parent AlB2 structure type is disfavored in Au-rich compounds due to soft atomic motions along the c axis. The instability of AlB2-type LaAuGe is confirmed by the presence of imaginary modes in the phonon band structure that correspond to a "puckering" of the hexagonal AlB2-type lattice, resulting in the experimentally observed LiGaGe structure type. The impact of size effects is less clear for Au-poor compositions; instead, twisting the AlB2 structure type to form the ThSi2 type opens a pseudogap at the Fermi level in the electronic density of states. This investigation demonstrates how crystal structure in solid-state materials can be compositionally tuned based on balancing size and electronics when multiple structure types are in close thermodynamic competition.