the only way I see this working is if you set up the security people as a privileged class so they aren't incentivized to side with the common workers to overthrow you. there's probably a minimum scale (ie, number of people) needed for this to be stable. it would still be hard to prevent the security from realizing they could just kill you and add your share to their own.

in a weird way, an (ex)billionaire commune might be the only viable bunker society.

So the bunker could help survive the initial strikes of a nuclear war, assuming your country can quickly retaliate and end it before substantial or permanent damage is caused. Then again in a country as big as the US the chances of a nuclear warhead penetrating the defenses and landing close enough to be a real and immediate danger are relatively low.

But if we're talking about prolonged nuclear war, anything that would really upend social order, or actually endanger the survival of the species (nuclear winter), such a bunker is just a fancy coffin. They will never get the chance to get out of it. If they do there will either be scorched earth waiting for them, or some sort of society that would see the protected elites as enemies and a cache of resources