The Air Force has an initiative called Kessel Run [1] to embrace agile software development — that might be your best route to building better software for the DoD. I'm not sure if other branches have similar programs.
https://breakingdefense.com/2021/10/not-going-solo-air-force...
Other responses have mentioned some routes inside... but beware... make the incumbents look bad too much and you'll be bought out and shut down.
Pride takes a seriously distant back seat to maintaining the current procurement monopolies and relationships, and maximizing contract payments for the least effort.
I'll stop there before I start to incriminate myself or get on watch lists if I mention possible solutions.
So that means you'd basically have to do business development first with an organization like NUWC Newport or NUWC Keyport (the undersea warfare centers who do submarine R&D) to drum up interest so that the government can issue an actual RFP or similar, and then you could bid to work on it.
I'll be the first to admit it's super difficult but if you're interested you might also consider talking with a local NavalX Tech Bridge (see https://www.secnav.navy.mil/agility/Pages/tech_bridge.aspx)