>You cannot know if a bloody invasion was the only other option.
Well that was the shovel ready option that they were actively preparing for. IIRC it was scheduled for November so unless some other unexpected circumstance was going to force the Japanese into surrendering before then the invasion was going to happen.
>Or bombing a non densely populated area?
Considered and ruled out in 1945
>Also, you simply can't count the causalities of nuclear bombs the same way as conventional ones
Yes you can. It's done the same way. Start at zero and go up.
>since there will be related deaths for generations; increased cancer rates, birth defects etc.
This is only an issue for the people who survived the bomb first hand. It's not like their grandkids are going to suffer radiation sickness.
I'm not sure if I'm misinterpreting your tone but your comment reeks of being in an ivory tower and out of touch. There was every indication (Iwo Jima and Okinawa to name two) that the Japanese were going to fight to the last man on the home islands. There was a Japanese army running around beheading everyone in China. The American blockade was fighting off kamikaze attacks regularly. They could have firebombed those cities if they wanted but that wouldn't have sent the same message to Japanese leadership.
Every day the war continued cost lives. Using atomic weapons and ending the war sooner was well worth the extra cancer. I seriously cannot fathom the kind of ideology or belief system that leads one to conclude otherwise.