Yeah, even ignoring the politics, Starship should be able to get launch cost <$200/kg, lower than just the energy costs for the space elevator in Edwards' study. You're not going to get launching something with a nuclear fission bomb and a massively hardened nuclear space gun to be lower than that, just due to the costs of the bomb itself, never the massive propellant costs and complications of snagging it from a suborbital trajectory with a Starship.

Just put the payload on a Starship.