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I am using ArcGIS Pro 2.3.1.

I have an elevation raster in WGS84 and a national Z coordinate system. I need to project this DEM raster to the EGM96 Z coordinate system.

Which XY coordinate system goes with that?

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Your data is in a compound coordinate system; i.e. vertical data in one coordinate system and horizontal in another. This is true both before and after transforming the raster to EGM96-relative heights.

The horizontal (XY) coordinate system can be anything you choose - though typically one would select a map projection tailored to the area of interest (such as a national grid or a UTM projection). Alternatively, you can keep the horizontal component in WGS84.

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