As a former mentor of then Senator Harry Truman upon his arrival in Washington; FDR's Assistant President for much of WWII; a pre-convention favorite in 1944 to be FDR's running mate; and newly appointed Secretary of State; Jams Byrnes certainly wielded considerable influence with Truman during those summer months of 1945. However it is worthwhile to remember the slogan over Truman's desk for his entire presidency: "The Buck Stops Here". Truman accepted full responsibility for every decision from the Oval Office, and expected to make the final decision on all key government policies.
However, the nature of the allegations made concerning Byrnes' involvement in the Hiroshima and Nagasaki decisions seem to involve more manipulation of Truman than influencing. The distinction I make between these terms is that influencing involves the presentation of all available knowledge accompanied with a strong but balanced argument in favour of a preferred decision; manipulation involves the presentation of only knowledge and evidence in support of the preferred decision, resulting in a biased and unbalanced argument being presented. Byrnes' preference for manipulation rather than influencing of Truman seems to have been the cause of their falling out in 1947. In this light, and given the facts noted in papers such as this one, my conclusion is that Byrnes' engaged in much more manipulation of Truman than influencing of him; but that given his personality, Truman would never-the-less stick by his decision.
Byrnes is noted often for the copiousness and precision of his conference notes, but any notes he may have made on the Atomic Bomb decisions in July and August 1945 seem either not to exist anymore, or to still be classified. As such, we are left with only the published personal recollections of those present during those meetings, always remembering the perils of taking too literally the memories of those who are "remembering with advantages".