The Law of Non-Contradiction has nothing to do with proving a negative

The law of Law of Non-Contradiction (LNC) is defined as follows:
"It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect, and all other specifications that might be made, let them be added to meet local objections ." It will be noted that this statement of the LNC is an explicitly modal claim about the incompatibility of opposed properties applying to the same object. It does not apply to proving that something cannot exist. The problem with proving a negative is that it is impossible to know that something does not exist somewhere.

As Aristotle himself stated, “a demonstration of everything is impossible”, resulting in infinite regress.

So, what it gets down to is that to prove a negative, you have know everything there is to know in the universe.
This is really the idea behind the claim that "you can't prove a negative"--that we don't have the resources or ability to exhaustively enumerate all examples over the entire universe. For example, Betrand Russell postulated a teapot orbiting the sun - how would you prove it wasn't there?