It was not an either/or decision. It was never weighed as a "do we drop the bomb or do we invade?" question. The only real question on the bomb was, where do we drop it, and when? If it ended the war, great. If not, they were still planning to invade.

The idea that there was a long, carefully considered "decision" about using the bomb is a postwar construction that does not adequately capture how they were thinking at the time. To use the bomb was the "straightforward" choice given all of the decisions that had already been made regarding making it, readying it for combat use, etc. To not use the bomb would have been the hard choice, the real "decision" — and nobody was really lobbying for that. Certainly not Truman, who doesn't seem to have given the matter much thought at all. He liked the bomb because it could potentially end the war before the Soviets got involved. Invasion was still a long way off (three months).

This is not to say that there were not lots of decisions made at the end of the war — things like, "what terms of surrender should be demanded of the Japanese?", for example. But "should we use the bomb or should we invade?" wasn't one of them.

In the postwar, a narrative about the "decision" was created in part to justify having used the bomb. But it clearly overstates the amount of discussion, debate, and deciding that went into the use of the bomb. As the historian Stan Norris has put it, "Truman's 'decision' was a decision not to override previous plans to use the bomb." It is not clear that Truman regarded the bomb as something that needed special decision-making to use (they had, after all, already been targeting Japanese civilians with incendiary bombing), and there is no 1945 evidence that shows that Truman was ever shown high invasion casualty projections and asked to make any kind of "choice."

For a good walk-through of the decisions that faced American, Soviet, and Japanese high command at the end of the war, see Hasegawa's Racing the Enemy.