This is not quite correct. LLNL is a Federally Funded Research & Development Center (FFRDC) which is owned, as a facility, by the government, but managed and staffed by a non-profit contracting organization called Lawrence Livermore National Security, LLC (LLNS) under a contract funded by DOE/NNSA. The board of LLNS is made up of representatives from universities (California + TAMU), other scientific non-profits (Battelle Memorial Institute), and private nuclear ventures (e.g. Bechtel.) LLNS pays, with very few exceptions, staff salaries at LLNL, and they are not beholden to the government civilian pay schedule.

https://www.llnl.gov/about/management-sponsors