Trying to justify ticketing fraud, e.g. transporting your pet as an ESA, because you don't like the fare charged is higher than you personally wish to pay is exactly that: Fraud. When WN catches a fraud, it ought to ban the passenger.
Even the new DL rules and most certainly WN's feeble rules do not come close to what DOT authorizes carriers to require and which carriers ought to require.
The DOT rules permit the carrier to require advance notice, require medical certification from a treating professional of a specific DSM-recognized mental illness and reasonable certification of the animal's training. The second requirement is particularly important. While there seem to be plenty of people willing to generate online ESA letters, there are not that many sane people who want a medical diagnosis of a mental illness when they do not have one.