When looking to pass new laws or repeal old laws it can be useful to point out the experience in other States.
So far as I know, Utah has never had a law nor rules banning the possession of guns in our State capital. Certainly for the last 20 years that I've been involved in the RKBA community here, no such laws nor rules. We have several members of our legislature who are well known to CC daily. It is not at all unusual to see OC or very casual CC in our State capital including in the Senate and House galleries that look down on the floors of the respective chambers. These balconies are open down to the floor, no glass separations.
Utah does not have a State Police force per se. Rather, our Highway Patrol is responsible for security at the Capital and for protecting our governor, Lt. Gov. etc. Those assigned to the capital or to protective duty are well versed in the law and are utterly professional and courteous in their interactions with gun carriers. And as you'd expect, our gun carriers have been, without exception of which I'm aware, courteous and appropriate in their conduct.
I am not aware of any incidents at our capital involving either CC (requires permit) or OC. I'm trying to recall if there was a criminal incident a few years back, but can't find anything at the moment.
In brief, no restrictions to speak of at the Utah State capital and no issues with that in the last 20 years.
I wish you the best of luck in winning expanded statutory and rule recognition for your RKBA.
One thing we found very useful in Utah was for a pro-gun legislator to invite his colleagues (pro and anti-RKBA) to attend a Utah permit class that a couple of instructors would present to legislators and their spouses for free. Since our class focuses mostly on use of deadly force and other laws related to guns, it provides some very good instruction and reality for legislators, most of whom know very little of what gun laws actually say. I won't say we've turned any antis into allies, but we've sure seen a positive change in attitude toward RKBA in the legislature since this started.
Charles