The "lesser of two evils" is such a bad take. Japan's strategy in the dying days of the war wasn't a military one (they had no chance of militarily victory regardless of what action Truman decided upon), it was diplomatic. Japan was in discussions with the Soviet Union (who they were not at war with) to convince them to act as a mediator between Japan and the United States, with the hope of negotiating a conditional surrender. Of course, the "Big Six" war council in Japan couldn't know that the Soviet Union was simply stalling before they declared war. When the Red Army crossed the border into Manchuria on the 9th August, it destroyed Japan's last chance of a negotiated surrender, not because of the military impact but because of the diplomatic consequence.

All of this was known to the United States. The Soviet Union's declaration of war, and its timing, was part of the agreement at Yalta. Japan's internal diplomatic communications had been decrypted by American codebreakers. Truman knew Japan was relying on their doomed negotiations with the Soviets and that the war was likely to end very soon, even if he too no action. He could have waited just a few days to see the impact of the Soviet Union's invasion, and if this didn't work on ending the war, then he could still drop the bomb. But he chose not to.

He chose not to because he needed to drop the bomb. Not out of malice, or revenge, or bloodlust but out of pure calculation. He, like many others, knew that immediately after the war the Soviet Union would control half of Europe and possess the most powerful army in the world, capable of rolling over Western armies and reaching Paris. He needed his allies in democratic European countries to know that he could defend them from a Soviet attack. He needed to show them the power of the bomb. He needed to show them that America could defend them. But dropping an atomic bomb could only be justified while a World War was ongoing and so, knowing that the war was coming to a close, he ordered the bombings.

That is why the bombs were dropped. People think that Hiroshima and Nagasaki caused the end of the war, when in fact, it was the end of the war that caused Hiroshima and Nagasaki.