The Independent experiment with the Presidency unravels by 1984. The coalition that Anderson and Lucey had put together in 1980 and defeated an incompetent President was no longer viable. The Republicans had a strong candidate in the person of George H.W. Bush, former Congressman from Texas, former U.S. Ambassador to the UN, former Chief of the U.S. Liaison Office to the PRC, former Director of the CIA.
Bush is seen as more prepared by the electorate to become President than is the more untested Hart. Hart is more of a lightweight. As well, 1984 America would not likely have been prepared for a black Vice President. The Democrats were not about to nominate Jesse Jackson for President, nor was he picked for the ticket for Vice President IRL.
In 1980, you have Anderson winning 269 out of 538 electoral votes, not enough to win the Presidency outright, as he would have needed 270 to win outright. 269 votes would be exactly half, but not a majority. So you are assuming that the Democratic controlled House of Representatives of the 97th Congress of the time would have put the Independent Anderson into the Presidency and that the Republican controlled Senate of the time would have put the Independent Lucey into the Vice Presidency. I am not certain that Democrats in the House and Republicans in the Senate would turn over the Presidency and the Vice Presidency to two Independents. Although, since the Independents came so close to winning outright, they may have felt compelled to do so.
Of course, Bradley was never Governor of California, but I am assuming that you have made him Governor in this scenario in order to make him a more viable Vice Presidential nominee.