This question really depends on exactly which predators you're talking about. However, a lot of the predator fish are actually 100% coral safe. They're just listed as not reef safe because they will eat smaller fish or inverts, especially shrimp. I have 3 lionfish and a snowflake eel in a mixed reef tank without any issues. I actually really don't have any issues with the CUC either, but again that depends on what predators. Lionfish will eat shrimp, but mine show absolutely no interest in my snails, conchs, hermit crabs, mithrax crab, urchin, or serpent starfish. My snowflake eel also completely leaves my CUC alone. However, I know triggers will eat all your snails, so you have to get more creative with some of the predator fish in terms of CUC.
I have 3 different lionfish species and a snowflake eel with gorgonians, sponges, soft corals, zoas and palys, LPS, SPS, anemones, and tridacna clams without any issues. As others have mentioned, the big consideration with predators and corals is having enough filtration to support the bioload. I run a skimmer that is rated for twice the water volume of my system, a biopellet reactor, and have a refugium with macro algae. I also try to stick to weekly 10 gallon water changes to keep nutrients at a reasonable level. If you have enough filtration and keep up with water changes though, you can definitely have predator fish in a full mixed reef tank.
Here's my predator reef