In ArcGIS Pro you can do this easily. Just load your mosaic dataset to ArcGIS Pro and select the processing template that represents the hilllshade which will draw by default with the black to white (no transparency) color ramp.
If you go into the symbology of the image layer, you can choose the drop down list of color schemes, which will provide access to format the color scheme::
In the next dialog, you will be able to change the white color
...to white with 100% transparency:
... which will yield the following effect: