Invasive signal red crayfish are as you say non native, the EA do issue permits tho they are understandbly strict on the type of trap used as water voles, otters and our own native crayfish must not be trapped, just like you i want to do my bit to eat my way through the problem, but apparently its not so black and white, the signal red crayfish is a canibal that will happily eat its own young, so by catching a bag full of these buggers you are actually reducing the predation of the youngsters, thereby increasing the population doh!
Keep me up to date on your permit progress, as i gather they are brilliant boiled, then the meat fried in garlic butter, recipes welcome.