It is not that complex of an issue. Thousands were dying in very brutal fighting. And the closer we got to Japan, the more brutal the conflicts were becoming. No matter what we tried to inflict on the Japanese to get them to stop, they would not stop fighting.
Everyone deseperately wanted an end to the fighting and the misery and the personal tragedies being visited on thousands of families back home. We wanted peace and the only way to acheive that against an adversary that would not give up was to win the war.
A solution came in the form of a bomb, a very big new bomb. Not a lot was known about it except that it would be very big and would be horrible for those where it fell. And it was felt that this was the best chance for ending the war. We knew it would kill many, many people. But we also new that to get to the same conclusion - an end to the war - just as many would die in conventional bombings and fighting.
We had a decision to make. Do we drop this thing knowing it will kill many in an instant or do we continue to fight towards the same conclusion and have just as many die slowly? We made our decision.
We dropped one of two bombs. It was a very big blow to Japan. We then isssued communication to Japan asking them to surrender and stop fighting. We waited a week for their answer. Conventional fighting continued elsewhere with many thousands continuing to dye on the battlefied and the seas.
No word came from Japan so after a week we dropped a second bomb and again we waited. The answer came. Japan surrendered.
It does not seem like a complex set of circumstances to me. What is it that you find hard to sort through? I presume you do not think the bomb should have been dropped.
I don't mean to nitpick but the decision was even clearer and simpler than that. At the time we could fairly accurately predict just how many Allied soldier's deaths it would take to win using conventional warfare by looking at the battlefield reports at hand and weighing that against the known populations of the islands we were fighting on, the increasingly adverse (to us) terrain we would face and the demonstrated fact that they would fight to the last man, woman and child. We estimated an additional 100,000 Allied deaths.
What was also obvious from this was that we would have had to kill far more Japanese and Okinawans by an order of magnitude than the two bombs did (including all subsequent radiation sickness deaths.) One might naively argue 'how could we know that since we didn't know exactly what the bombs would do?'. Ignoring the fact that we did have a pretty fair idea what the bombs would do from tests, we knew for certain that one bomb could only kill all of the people in one city at best. Taking Japan and Okinawa by conventional means (hand to hand on the ground) would have meant killing virtually everyone in every city, every town and all rural areas to boot save an extremely small percentage.
How anyone can seriously question which choice was more compassionate strains my mind to know what the word compassion means.