The mountain in question is about the same height as the french alps. The headquarters in the should hold 500 people, is this possible?
The US did it 50 years ago.
In 1957, the Strategic Air Command began construction in New England inside Bare Mountain for a hardened bunker to contain the command post for the 8th Air Force, which was located at nearby Westover Air Force Base, Chicopee, Massachusetts. This underground facility was nicknamed "The Notch" (or formally as the 8th AF "Post-Attack Command and Control System Facility, Hadley") and was hardened to protect it from the effects of a nearby nuclear blast and designed so that the senior military staff could facilitate further military operations. Four years later, construction at Cheyenne Mountain was started to create a similar protection for the NORAD command post. Cheyenne Mountain was excavated under the supervision of the Army Corps of Engineers for the construction of the NORAD Combat Operations Center beginning on May 18, 1961,:18 by Utah Construction & Mining Company.
You can assume that HQ as surely possible from technology point. Cold War gave us a lot of bomb-shelters as well as top-officers HQ bunkers (mid of XX century) so nothing impossible here (just a question of budget). Also, you can take a closer look at Elon Musk's "Boring company" that plans to hollow long tunnels for high-speed commute lines.
The only serious enemy here is Earth itself with tectonic and volcanic activities, so rocks can be not that "eternal" like we tend to think.