Bridj, the on-demand private bus startup, said Thursday that it has raised $4 million in funding to help it expand beyond Greater Boston. The Cambridge company also said that it hired Gabe Klein, formerly a vice president at Zipcar, as its chief operating officer. Klein previously served as Chicago’s transportation department commissioner and as transportation department director in Washington, DC. The investment comes as Bridj expands its routes in Boston. But it may face regulation in Cambridge that could “essentially ban” the service in that city, founder Matt George said recently.