The billionaire is leading a group of investors in the unsolicited offer, which complicates the start-up’s plan to raise more money.

The nominee for F.B.I. director made his nonprofit into a publicity machine, selling his children’s book, his clothing brand and his image as Donald Trump’s ultimate loyalist.

Their lawsuits said the so-called Department of Government Efficiency, President Trump’s cost-cutting operation, violated a law requiring federal advisory committees to include a balance of views.

The New Georgia Project, whose leaders also included Raphael Warnock, now a senator, admitted violating campaign finance laws. It agreed to a $300,000 penalty.