There is nothing in the Constitution that demands that the loser of an American election has to personally congratulate the winner or even attend the inauguration. The transition will be carried out on January 20th when Joe Biden and Kamala Harris are inaugurated no matter what Donald Trump does in the remaining few months.
William Jennings Bryan, a deeply committed Christian, started the modern practice of personally conceding his loss to the winner in 1896 when he sent a letter to William McKinley. In every election since then, a personal concession has been carried out by the loser to the winner within a few days of the election being over and all of the losers have attended the inauguration in order to show the country that our democratic system is strong and the business of the government will continue unimpeded.
The only exception was in 2000, when Al Gore did not concede for about a month while the deciding votes in Florida in that extremely close election were being contested. When that was finalized, Gore conceded to George W. Bush and attended his inauguration. Gore was severely criticized by many Republicans for being a sore loser. And even the 9/11 Commission said delaying his concession could have hurt American foreign policy. But in that case, President Bill Clinton invited Bush and his team to the White House to read government documents and meet with the people they may replace within a few days of the election. He said the Bush team needed access as soon as possible for the good of the country because he might turn out to be the winner!!
The only election in United States history in which the results were not accepted at all was in 1860 when seven southern states refused to accept the victory of Abraham Lincoln and left the union, which initiated the Civil War. The country was torn apart and over 660,000 American lives were lost in the conflict. A hero of that war, Gen. Winfield Scott Hancock, later ran for president against James Garfield in 1880. He lost an extremely close election. He accepted defeat and against the advice of some advisors refused to even ask for a recount.
An authentic American patriot, Gen. Hancock issued this statement: "THE TRUE CHRISTIAN SPIRIT IS TO FORGIVE AND FORGET"!!! I say Amen!