Though perhaps it doesn't fulfill the conditions of the OP, I was toying with a scenario that had the South Carolinian battery on Morris Island fire upon, disable, and sink the Star of the West, a civilian ship sent to reinforce Fort Sumter on January 9th 1861. Whereas in OTL, only a handful of shots were fired and the ship retreated, in TTL that's off the table. The prolonged firing convinces Major Anderson to engage in counter-battery fire against Morris Island and Fort Moultrie. This in turn prompts the South Carolinian batteries to open fire on Fort Sumter. Perhaps there might be a way to de-escalate this conflict, but I'm skeptical. IMHO this engagement or a similar escalation of the reinforcement attempt has a good chance of destroying Buchanan's illusions of preventing a Civil War.
While I don't think Buchanan would call for volunteers to put down the rebellion right away, I think he likely begins the process of moving as much war materiel North as possible and reinforcing those forts best suited to withstanding attacks from secessionists. These policies and his earlier replacement of pro-Southern cabinet members, coupled with the increase in secessionist rhetoric prompts a faster secession process in the South. As slave states join South Carolina in outright rebellion, Buchanan is faced with no option other than calling for volunteers and bringing an end to the war. I'd wager that in this scenario a good amount of the war materiel that fell into Southern hands in OTL does not. For the sake of the OP, let's say that Buchanan's policies and appointments during the final days of his presidency prove to be good ones. When he's inaugurated, Lincoln inherits a Union that's well on it's way to winning the nascent civil war. Feeling more connected to politics and determined to see the war through in some capacity, Buchanan remains more politically active in his post-presidency career and is a staunch opponent of Pierce. Though he certainly wouldn't be the greatest president of all time in this scenario, it's hard to do worse.