Correct me if anything I say here is inaccurate or plain wrong but my understanding has always been that Japan's peace feelers had at some point successfully communicated to the Truman administration (well in advance of the atomic bombings) that they would indeed surrender on the condition that Hirohito be granted immunity from prosecution and allowed to remain emperor. By the end of it, Japan would surrender unconditionally and Truman chose to grant Hirohito immunity and the right to remain emperor anyway, despite heavily insinuating prior that they would do just the opposite. In general, was there any strategically justifiable reason for him to insist on unconditional surrender? Forgive the bit of editorializing I'm going to add here, and this isn't to spark a tangent or debate about the bombings but just to frame why the answer to this question is so important, but if it is the case that Truman had no reasonable cause to reject a conditional surrender would that not suggest that his subsequent decision to use the atomic bombs on populated cities, regardless of any other considerations that were subsequent to the Potsdam Declaration, was monstrous and cruel?
Yes, your understanding is more or less correct. (There are some nuances — they didn't ask for immunity, they basically just said that the Emperor would be exempt from the Potsdam conditions.)
Basically what Truman and his cabinet believed was that an unconditional surrender was necessary if they were going to rebuild Japan from the ground up. They did not yet have firm views on what they wanted to do with the Emperor anyway. Indeed, it was later decided that keeping the Emperor as a monarchical figurehead (and having him renounce his divinity) was the best choice for their aims during the Occupation of Japan, but that was later.
Truman himself rejected modifying the conditional surrender issue — including on this very point — even at Potsdam, despite urging from many advisors (including the Secretary of War, and Winston Churchill) that moderating the unconditional surrender requirement when it came to the Emperor would likely produce a swifter end of the war. Truman's essential argument was that Japan did not deserve anything less than unconditional defeat, on account of their perfidy at Pearl Harbor.
It is possible that the atomic bomb played a role in this; Truman made this decision not long after the successful news of the Trinity test, which by all accounts of those around him buoyed him greatly and made him very confident that he held a good hand when it came to the end of the war. In any event, Truman and his cabinet did not think Japan was in any position to negotiate, and that until they realized that, they would not really have surrendered.
In the postwar period, there was a lot of argument (interestingly, from conservatives) that the unconditional surrender requirement had unnecessarily prolonged the war. In the late Cold War, a similar argument would be made by those who believed the atomic bombs were unjust and avoidable. Either way, it's been a controversial thing for a very long time, for exactly the obvious reasons. Resolving it definitively — would modifying the surrender conditions have really lead to an earlier end of the war? — is necessarily impossible.
Separately, Truman did not make the decision to use atomic bombs on populated cities. He agreed with the plans in progress to use the atomic bomb, but it is not entirely clear he even understood the nature of the targets. That is a separate issue but I just wanted to point it out. The "decision to use the atomic bomb" narrative is essentially a postwar myth, made to make Truman seem more actively in charge than he really was; the atomic bombing was run nearly entirely by his advisors.