yemista,
Most people will tell you varying things, when I had my 5 reds, 3 of them usually stayed in a driftwood cave I had setup. Occasionally chasing each other around the tank. The 4th red made a nest on the other side of the tank and attacked any fish that got near. The 5th patrolled the tank constantly looking for food. He always hit the food / feeders before any of the other fish. This was the guy that would get just a little too close to my hand while cleaning the tank.
If you have a brave crayfish he will probably get eaten. If you have an active P, even if the crayfish is in a good hiding spot the p will find a way to get him.
I watched a pair of my reds trap a crayfish behind a piece of driftwood and they just waited for him to make a move. I found half of him a few hours later in the tank.
I have read that reds are less aggressive solo, so you may have more luck. Crayfish are easy to catch and are free food as long as you quarantine them. If they Co-hab then thats even better.