The strongest evidence against the teapot orbiting Neptune is the weakness of the argument against it. The argument against the teapot orbiting Neptune is there's no good reason to believe it exists. That's a weak argument, but sufficient to counter the unsupported claim that it exists.

We're talking about extraordinary claims, Bigfoot and the teapot orbiting Neptune. If I claim I own a stapler, there's are several inductive argument supporting that claim. One is that it's an ordinary claim, and usually when people make ordinary claims those claims are accurate. And most people own staplers, so it's likely I own a stapler.

But those arguments aren't sufficient to support extraordinary claims. If I claim doctorphyco is a murderer, the argument that usually when people who make ordinary claims those claims are accurate doesn't apply. And most people aren't murderers, so it's not likely doctorphyco is a murderer simply by virtue of being a person. Nobody should believe doctorphyco is a murderer just because I say so, even if I pass the look-'em-in-the-eye test and have a firm handshake.

Likewise nobody should believe I've seen a Bigfoot simply because I say so.

The argument for Bigfoot is premised on testimony. See above. The argument against Bigfoot is premised on the absence of a specimen.

Absence of evidence IS evidence of absence. There were 10,000 reported sightings of mountain lions in the eastern US between 1970 and 1990, but the number of cougar road kills didn't correspond to the road kill count in regions where breeding populations of cougar were known to exist. The lack of road kills indicated there weren't breeding populations of cougars in the East, but the testimony indicated there was. The lack of road kills trumps the testimony, because the testimony could be nonfactual, but the lack of road kills was certainly a fact.

When facts contradict testimony, fact prevails.