The newly discovered BaPt$_2$As$_2$ shows a structural distortion at around 275~K, followed by the emergence of superconductivity at lower temperatures. Here we identify the presence of charge density wave (CDW) order at room temperature and ambient pressure using single crystal x-ray diffraction, with both a superlattice and an incommensurate modulation, where there is a change of the superlattice structure below $simeq$ 275~K. Upon applying pressure, BaPt$_2$As$_2$ shows a rich temperature-pressure phase diagram with multiple pressure-induced transitions at high temperatures, the emergence or disappearance of which are correlated with sudden changes in the superconducting transition temperature $T_c$. These findings demonstrate that BaPt$_2$As$_2$ is a promising new system for studying competing interactions and the relationship between high-temperature electronic instabilities and superconductivity.
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