This is going to be a lengthy explanation, but I think to really understand why the government can do what it does, you have to understand the full scope of what a public forum is and what it isn’t.
A public forum is place where people are able to express their First Amendment rights. Public forums are generally split into two groups: traditional versus designated forums (btw, these aren’t legal terms of art). The traditional forum is one where there has been a history of people being able to use as a place to express their 1st Amendment rights. The most basic example of this is a sidewalk.
On the other hand, a designated forum is a place that the government has specifically set up for the purpose of free speech. This sort of forum has not traditionally been used as a forum for expression. A basic example of this would be a city council meeting room where they allow certain groups to use or certain topics to be discussed.
Finally, we have the non-public forum. This is a place that is not designated as a forum for free speech nor is it in the category of traditional public forums. Examples of this include public schools, military bases and governmental offices. Here, the government may restrict only the content (topics like abortion) of free speech, but cannot restrict the viewpoint (think either pro/against abortion) of the speech.
You’ve probably already figured out where state capitol offices fall. As an office space, the governor’s office is a non-public forum. The government never designated his office as a place for free speech nor is it a traditional place where citizens may voice their opinions. Thus, the government may restrict access.
Here’s how the case will probably play out: the prosecutor will allege that the protestors were trespassing. My understanding of Hawaii’s criminal trespass statute is that ”[a] person commits the offense of simple trespass if the person knowingly enters or remains unlawfully in or upon premises.” In this case, I’d say the protestors in your question probably knew they were someplace they were not supposed to be. The protestors would counter w/the free speech argument, then the prosecutors would rebut with the non-public forum argument.