Is there a way to detect if a cat is sleeping? 1.16

Solution 1:

There is no NBT tag to detect a sleeping cat. Why is that? NBT is not a format that is constantly manipulated during gameplay. It's just save data, the data that Minecraft uses to remember your world state the next time you open your world.

The only things that are stored in NBT is stuff that needs to be remembered if you close and reopen your world. Sitting and standing is one of those properties, but sleeping is not. Think about it a little. Cats only sleep when you're sleeping, and you can't leave the game without leaving the bed. And when you leave the bed, the cat no longer sleeps. Therefore, the cat is guaranteed not to be sleeping when you leave your bed, meaning that this data does not need to be stored in NBT, and hence, never is.

So it appears you're out of luck. I may be able to come up with a workaround, but I haven't yet, so I'm delivering this partial answer to you.