Nothing you're doing wrong. When it comes to projectors, they have limited light output as compared to a decent 4K HDR display. This is a problem because HDR needs lots of brightness (as in light output) to look good and perform as intended. Many current TVs are able to output 1,000 nits (or more) of light, which allows for good HDR performance. While your projector can "technically" receive 4K HDR, it likely lacks the capability to properly take that information and utilize it in a way to project an image that approximates a director's intent.
Here's where good mapping enters the picture. Tone mapping is a process that maps one set of colors to another to compensate for a display's limited dynamic range.
Your Sony 4K disc player and Apple TV aren't going to help you in this department. But, a Panasonic UB820 4K player (or the step-up UB9000) has powerful tone mapping abilities that take care of the necessary adjustments
before the signal is sent to your projector. You can buy either one (820 is about 1/2 the cost), make the necessary adjustments in the settings menu of the disc player, and you should end up with 4K HDR disc playback that's significantly better.
I'm tagging
@Robert Zohn and Steve
@mechman, because I think they'll probably have some insights or suggestions to add!