I would NOT expect to be harassed. It is perfectly legal to carry inside the park itself - there are no additional regulations inside a National Park. It is against Federal laws for you to possess a firearm inside a building in which Federal employees (Park Rangers and other National Park Service employees) regularly perform their duties. Such buildings should be posted with a sign required by 18 USC 930. If you do not possess a Wyoming recognized permit, it is still illegal for you to conceal a firearm on your person, because "constitutional" concealed carry on applies to persons eligible for a Wyoming permit, and Wyoming does not issue permits to non residents.
There is an almost never enforced Federal law that makes it illegal for you to possess a loaded firearm on public property within 1000' of the premises of a school, unless you possess a permit/license from the same state the school zone is located in (with a few other exceptions).