The Ken Burns 14-hour marathon wraps up tonight, well worth my time (since I've written several books about 20th century history and politics) despite some amazing oversights, such as only 80 seconds on Japanese internment during World War II.
Unfortunately, there's not much evidence either way but that shouldn't stop the speculation.
I don't have time to do a full treatment, but let's just say that the most brilliant probing of Truman and his motivation (obvious or hidden) for targeting 150,000 to 200,000 civilians for death in the two cities was found in, ahem, the 1995 book I wrote with Robert Jay Lifton--courtesy mainly of Lifton, I hasten to add--Hiroshima in America. Lifton spent about three pages on FDR and pointed out that Roosevelt, unlike Truman, had sought advice or discussed alternatives to using the bomb, so at least that was on his mind, if only in the back of it. Also, he was far stronger than Truman in power and confidence and would have been far more able to withstand the urgings of aides and generals. Einstein said he didn't think FDR would have used it. McGeorge Bundy and atomic scientist Phil Morrison said they agreed, in talking to Lifton. Most would dispute that.
Read about Leo Szilard's efforts to get Truman to hold off and ask yourself if FDR's White House would have reacted with more interest. And then there's the claim that Truman did not fully understand the civilian toll of using the weapon--would Roosevelt have been more in tune? And see how Truman opened the nuclear era with a lie. Would FDR have done the same--and gone on to speed, rather than halt, the chances for a nuclear arms race?
On the other hand, FDR had a lot invested in the bomb and would have had to defend the enormous resources poured into the project if he did not use it. But he also might have recognized, better than Truman, that Japan was basically defeated and would have likely surrendered in the same time frame due to the Soviets declaring war.
Let the debate continue. I just don't expect Burns to contribute to it.