Sometimes in doc shooting, you need to track a subject and Sony autofocus supposedly does a great job with that. Try gyro can help but your NLE, at least for processing that, has to be Resolve.
Sony just released the FX30 (APS-C equivalent of the FX3) and the body only is just $1800.
Again, it depends on what and how you shoot but I find the BMPCC 4K has limits when I do run and gun. It’s just not that convenient and I can’t risk losing unplanned shots.
Sony does overstate their dynamic range when you consider noise though so I consider them similar. Their color science has improved. BMPCC has the edge for planned shots, shots where you have a few seconds to set up but too much of my shooting doesn’t allow for that (ENG style and sometimes with equally fast turnarounds).
Camera prices are so low now it’s possible to own both although the lens is the cost both allow for Canon lenses as a middle ground.