Self-assembled InAs quantum dots (QDs), which have long hole-spin coherence times and are amenable to optical control schemes, have long been explored as building blocks for qubit architectures. One such design consists of vertically stacking two QDs to create a QD molecule (QDM) and using the spin-mixing properties of "moleculelike" coupled hole states for all-optical qubit manipulation. In this paper, the first of two papers, we introduce the incorporation of dilute GaBixAs1-x alloys in the barrier region between the two dots. GaBixAs1-x is expected to increase the spin mixing of the molecular states needed for qubit operations by raising the barrier valence-band edge and spin-orbit splitting. Using an atomistic tight-binding model, we compute the properties of GaBixAs1-x and the modification of hole states that arise when the alloy is used in the barrier of an InAs QDM. An atomistic treatment is necessary to correctly capture nontraditional alloy effects such as the band-anticrossing valence band. It also allows for the study of configurational variances and clustering effects of the alloy. We find that in InAs QDMs with a GaBiAs interdot barrier, electron states are not strongly affected by the inclusion of Bi. However, hole states are much more sensitive to the presence and configuration of Bi in the barriers. By independently studying the alloy-induced strain and electronic scattering off Bi and As orbitals, we conclude that an initial increase in QDM hole-state energy at low Bi concentration is caused by the alloy-induced strain. We further find that the decrease in QDM hole energy at higher Bi concentrations can only be explained when both alloy strain and orbital effects are considered. In our second paper, we use the understanding developed here to discuss how the alloyed barriers contribute to enhancement in hole spin-mixing and the implications for QDM qubit architectures.