I think if you did it without a federal depredation permit it would put you in conflict with the Endangered Species Act and subject to federal prosecution.
That is the same message that I have been giving for several years. The question went to the highest levels of the USFWS and the answer that came back was that they would investigate any cougar killing in TN of which they became aware. If there was no evidence it was an escaped or released "captive" cougar, that they would seek prosecution under the ESA in federal court.
Shooting one seems like a bad idea to me.