It's more likely they exist, and the government isn't suppressing anything or even think they exist.
I hate to keep repeating this but most of the giant government we have doesn't care at all about Bigfoot, swimming salmon, or Asbestos in my basement. The government is big because the nation is big and thus its needs are large and complex. It gets bloated the same way a giant corporation in the private sector can get bloated as it grows.
There could be a tiny part of the government who might be in the business (has a mission) to care about Bigfoot or whatever. Nearly 99.99999% of it could care less.
Assuming that .000001% cares, they either think/know bigfoot exists or thinks it could exist. Either way, I just don't see a motivation to suppress Bigfoot knowledge. Once a bigfoot is killed then various parts of our large government would wake up in the areas, they concern themselves with. Some tiny part of the government might then enact protections under some wildlife act. Others would alert the coroner of the country. Still others might pass laws making hunting a Bigfoot illegal. Various parts of the government machine might be expected to kick into gear as well in response to the body. But such a government is not going to cut my break lines if I had proof bigfoot exists to silence me. I'm not saying our spies wouldn't do this if needed. Just not for bigfoot. Nuke secrets in 1945, yes.
Really ask yourself if the significance of keeping bigfoot quiet is large enough to warrant a suppression campaign or worse yet, killing people.
Power brokers in government can be "bribed" to adopt various positions. But generally, most guidelines and positions come from scientific consensus. Like it or not at this time the scientific consensus is Bigfoot doesn't exist or at least has not been proven to exist. That is not a radical Idea. That is reasonable under the circumstances. The parts of the government which might be concerned about Bigfoot should it exist right now don't care. The other parts don't concern themselves with it at all.