We present a detailed experimental and computational investigation of the influence of pressure on the mixed-anion oxyhydride phase Ba2YHO3, which has recently been shown to support hydride conductivity. The unique feature of this layered perovskite is that the oxide and hydride anions are segregated into distinct regions of the unit cell, in contrast to the disordered arrangement in closely related Ba2ScHO3. Density functional theory (DFT) calculations reveal that the application of pressure drives two sequential B1-B2 transitions in the interlayer regions from rock salt to CsCl-type ordering, one in the hydride-rich layer at approximately 10 GPa and another in the oxide-rich layer at 35-40 GPa. To verify the theoretical predictions, we experimentally observe the structural transition at 10 GPa using high-pressure X-ray diffraction (XRD), but the details of the structure cannot be solved due to peak broadening of the XRD patterns. We use DFT to explore the structural impact of pressure on the atomic scale and show how the pressure-dependent properties can be understood in terms of simple electrostatic engineering.