I've never had problems with survival of alien crayfish - quite the reverse in fact. I can imagine a severe change of temperature might kill them off, or lack of oxygen if they're overcrowded and the water was not being refreshed through the night.
I think I'd want them to have plenty of fresh water before cooking them, depending on where they were trapped (I've only ever taken them from gloopy stillwaters and perhaps a clear free-flowing upper Thames is OK).
I had no idea it was illegal to trap them. I thought it was positively encouraged. There was a chap some years ago (I don't know if he's still around) who used to trap them in the Serpentine and serve them up in his cafe on the Edgware Road.