Does it matter? It was probably obvious to the scientists working on the bomb that other countries would get it too sooner or later, including countries at odds with each other.
Could Hirohito (Suzuki, etc.) have been convinced by bombs dropped elsewhere?
(our physicists were able to back-of-the-envelope; should their physicists have needed hundreds of thousands of civilian deaths to calculate what an A-bomb could do?)
So I think it's not obvious (multiple books have been written on the subject) what could or should have been done or not done back then; now, from my point of view, those cards have been dealt, for good or for ill.
Hirohito was apparently convinced after the bomb on Hiroshima, the cabinet and military staff still wanted to fight on after the bomb on Nagasaki. They even tried to block his radio speech.
Personally, I think it was tragic, but there was not much choice. Forcing Japan to its knees by conventional means would have been a prolonged bloodbath (with the Soviets getting in the game as well), with probably a higher death toll.
But was it really obvious? From what I can tell, much of the nuclear arms race happened thanks to espionage. Had information on warheads and the like been properly contained, maybe other countries would not have so easily developed the bomb.
Not so easily, but they would have done it. Once it was known, there wouldn't have been any way to stop Stalin. His paranoia knew no limits. And then there would have been dozens, hundreds when the war would break out, nobody would be scared to use them.
Unfortunately, democracies have their downsides. One of them is that "properly containing" things is extremely hard. Soviet spy network was vast and infiltrated all tiers of US society, and without instituting some extremely draconian policies, compared to which McCarthy would look like a hippie, I don't see how it would be possible to prevent it. Especially given a lot of top scientists were, unfortunately, if not communists themselves than quite friendly with communists and would probably refuse to cooperate with any draconian regime that would be capable of suppressing the communist networks.
Japan was better off in the long run being occupied solely by the US instead of a split occupation with the Soviets like Germany. If we hadn’t dropped the two bombs, the Soviets were set to invade northern Japan.
They were allies by necessity, but I don't think there were a lot of illusions about where things age heading. After all, the official doctrine of the Communist Party has always been that every non-communist regime has to be violently overthrown and replaced by a communist one. USSR didn't have the bomb not because they didn't want one, but because they were incapable of building one by themselves, and stealing all the details by means of vast spy network they had in the US, and then recreating them on their side, took time. If they had the capacity, they'd do it as fast as possible.
Lagniappe: https://www.youtube.com/watch?v=oRLON3ddZIw#t=15s
[0] first test: 29.08.1949
[1] a year in which the US and USSR were, however tenuously, still allied