The problem with the trail camera is that the chickens in the house would trigger it constantly with their movements, and I would have to plow through thousands of pictures or hours of videos to catch the snake in the act. On the other hand, if it is snakes that are stealing fake eggs... Let them! I am sure, as you wrote, that they will die of indigestion. Those eggs will not be digested and will not pass through the snake. I have a few more of those eggs. I am going to place them in the nests to soak in the smell of chickens and eggs...
One thing you could try (please don’t quote me on this, I heard it from my neighbor), is to take a smallish egg, and work a large fish hook (catfish hook?) into it with the hook tied to line or string. Tie the string to something stable (a cinderblock might work since it has a hole in the middle). It shouldn’t harm the chickens at all as long as the point of the hook is inside of the egg since the chickens won’t swallow the egg whole.
The shake (if that’s what it is) will swallow the egg, which has a hook tied to a string in it, so the string will hang out of its mouth. Once he gets it down, he probably won’t want to give up the egg. Then as he starts to digest it, the hook will become exposed and will become caught in his stomach. Then he can’t leave.
Check multiple times a day to see if you ‘caught’ anything so you can kill it quickly. (I wouldn’t want a snake to suffer for too long)