You injected a bit more than just the belief held up against another belief in your question. You added the weight of one of them already having been accepted and a measurable risk hinging on whether or not it is even true. When we're evaluating the two propositions, stand-alone, without such high-stakes already in play, then yes, they are equally unfalsifiable. I don't have good evidence that literally demonstrates that Godzilla doesn't exist, just as someone attempting to evidence his existence would have poor evidence also. The prudent thing to do in these situations is to simply await further demonstration either way. If such never arrives, then you have taken no action based on potentially false information. And our having zero good evidence of Godzilla's existence literally equates to Godzilla posing no threat to us... so we can safely act and react as if he does not exist... until we can't.
In your scenario, people with very poor evidence already concluded that Godzilla does (or is likely to) exist and spent millions of dollars. And such things, unfortunately, likely are possible with more plausible subject matter. For instance, having poor evidence that "weapons of mass destruction" are located in a particular area, but proceeding with a series of tactical extraction/destruction missions anyway. Obviously the risk is going to be weighed based on the plausibility of the article under consideration, but it is not unheard of that action is taken based on poor evidence/information.
So, in the case of the "weapons of mass destruction" business - which has more weight? The belief that there are WMDs in a given area, or that there are not WMDs present? Wouldn't you have to say that the weight to be attributed depends on the evidence that there is in support of either case? Meaning you have to have at least some compelling evidence either way before one belief-option seems weightier than the other.
Bathos Logos said: "I don't have good evidence that literally demonstrates that Godzilla doesn't exist." Clara Tea responds: Well, I discussed the matter with Godzilla, and Godzilla said "AAAARRRRRR," which I think means that "I don't exist" (in dinosaur language).
Many knights have told kings that they slayed dragons. Given the fossils of huge dinosaurs, it sounded reasonable that huge creatures might exist, and if they insist that they spew fire, then that, too, sounded reasonable.
Bathos Logos said: "poor evidence that "weapons of mass destruction" are located in a particular area, but proceeding with a series of tactical extraction/destruction missions anyway." Clara Tea responds: (Ignoring Revelation that warns us not to attack Iraq or face God's wrath, such as Revelation 15 (seven plagues), and ignoring God's commandment not to kill). But, you are right, just as humanitarians, without religion, we should have had the patience to not attack Iraq. The reasons for war were laid out in the War Power Resolution of 1998, signed by both houses of Congress (both were Republican majority), and signed by President Bill Clinton (Democrat). Yet, Republicans didn't want Clinton to appear powerful or decisive so the resolution prohibited war, and merely gave Clinton authority to sow dissention in Iraq.
The 911 attack told us that somewhere in the Middle East there were terrorists (suspected al Qaeda at the time), and Osama bin Laden was said to be the "Where's Waldo" of terrorism by the FBI, who claimed that wherever terrorists met, there was bin Laden, somewhere in the picture. So, prior to the war in Iraq, we had nothing but a pile of innuendos. But, the America people didn't have access to covert intel. We were supposed to trust our president, and trust that the intel was sufficient. W. Bush and Colin Powell outright lied to the American people because they felt that keeping Americans safe was more important that the huge loss of lives in the Middle East.
As W. Bush said that nuclear bombs are not off the table (that is, he might nuke Iraq), the Doomsday Clock people refused to advance the clock (that is an estimation of how close we are to a nuclear war). I wrote to them, and they wrote back, saying that they didn't want to deter W. Bush from his war in Iraq. So, it depends on which political party is in power, not how close we are to a nuclear war.
Iraq (population of 30,000,000 lost about 1,000,000 citizens). Many of the soldiers sent to the front lines were shopkeepers, not soldiers, and were told that if they turned back they'd be shot. The sanctions against Saddam resulted in the starvation of Iraqi citizens (kids and women among them), and didn't impact Saddam at all, who was later found with a billion dollars in US currency in an oil drum, and Saddam, all the while, was busy building more palaces while his people starved (they were grateful for the employment). So, virtually every Iraqi citizen knew a friend or coworker who was killed by the US (that's no way to make friends). Now, after losing a war of 30 years, the US faces great resentment for their war.
This is from a region that still resents the invasion of Alexander the Great (they hold a grudge for a long time).
You made a good point that we must constantly form beliefs about the world (do they have terrorists, do they have WMD, how confident are we about our intel), and sometimes we have to quickly act on scant evidence (attack them before they attack us). So, our belief system not only establishes our theology, but also the political climate of the world.