If this unlikely scenario takes place, he would be best to wait until after the inauguration. The Constitution is murky on what happens if a President elect refuses to take office. So, if Biden were to resign, Kamala would immediately become president but what happens before or even the electoral college votes is kind of up in the air.
Jeff Greenfield wrote a novel about such a scenario, back in the 90s. In it, a presidential candidate picks a VP who is pretty different from him, ideologically, in order to pick up some southern states. He wins. Then, in a post election celebration, he is thrown from a horse and dies. The people who supported the president aren't sure they want the VP as president and insist they can nominate someone new to take his place. States where the VP-elect is popular are adamant about his being sworn in. The opposing party sees an opportunity to throw things to the House, where they would win. Alternate slates of electors are chosen, National Guards are called out... and it looks to be going to the SCOTUS because the constitution says NOTHING about what should happen. Interesting novel that only became more chilling after the 2000 and 2020 elections.
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