They probably would not even bid. Working on an OS project is fundamentally providing labor hours. Not high margin, no lock in, no investment and high profit tail on the business. It would end up going to little companies providing bodies at a low labor rate.
No way if those companies had to OS their code.
You could pump money into OS slowly by making government departments pay a royalty to the maintainers/projects they work with. Things such as Drupal, Tomcat etc.
I’d expect companies like Raytheon, Cerner, Lockheed Martin, Boeing, and HPE/CSC/DXC to win a supermajority of those contracts.