The best chance of avoiding Fort Sumter would have been for the garrison to have stayed in their original indefensible location on the mainland rather than sneaking out to the more defensible Fort Sumter. That could have happened either through more vigilant or lucky confederate militias catching the troops early in the operation, or through a different federal commander who adhered more closely to the letter of his orders or simply didn't make a decision like a few other commanders did. If the Federal troops had remained on the mainland, they would have probably been forced to surrender in fairly short order, with no chance of resupply, especially if the confederates took over the initially unmanned (except for a few workmen) Fort Sumter. That's actually another possibility: the confederates take over Fort Sumter before the federal garrison moves there. Yet another possibility: Fort Sumter was actually built by the South Carolina state government, then turned over to the feds. I suppose South Carolina could have hung onto it, or turned it into something other than a fort, in which case it wouldn't have become a flashpoint
If the garrison at Fort Sumter had not been a flashpoint, there were other potential flashpoints, most notably a federal garrison in Florida. Unlike Fort Sumter though, the Florida garrison didn't dominate one of the biggest southern harbors and a major southern city. The south might have been content to let it sit with confederate forces watching it.
As to whether or not the confederates could have been allowed to go without a fight: That's a tough one. Lincoln obviously wouldn't want that outcome. At the same time, there was some sentiment in the North to "Let the Wayward Sisters Go." There was also widespread sentiment in the border states against bringing the seceded states back by military force. Lincoln had to be careful not to appear to be the initiator of a war on fellow Americans.
Historically, Lincoln was much more canny than the Confederate leadership. He maneuvered them into a position where they were likely to fire first, though they didn't absolutely have to. The confederates made their position worse after Fort Sumter by commissioning privateers to attack Union ships.
Probably the more interesting question is: could a more politically canny confederate leadership have been able to make it politically impossible or politically not worth it for the Union to attack? I'm not sure about that one. They might have pushed harder to get Federal garrisons out during the 5 months of Buchanan administration that happened after Lincoln got elected. They could have pushed for that on the grounds that the garrisons were a cause of ongoing friction and that they simply couldn't guarantee their safety. They could have urged southerners in the various branches of the federal government to remain at their posts until the feds recognized the confederacy, especially Senators and Congressmen. That would have given Lincoln an interesting dilemma: there is no basis for removing a duly elected Senator or Representative, but keeping those guys in Congress would be absurd, and it would make the war effort much more difficult. Good luck on getting war appropriations past a southern filibuster.
President Buchanan shares a lot of the blame for the Civil War. He tried to kick the can down the road to the next administration, without making the tough decisions that needed to be made. A more forceful president could have probably prevented secession from developing much of a head of steam. A state legislature meets to vote to secede? The federal government calls up militia units from loyal states and moves them into the state, maybe a few miles from the state capitol, maybe in the streets outside the legislative chambers. If they vote to secede, you arrest them for treason. Not sure if that would work. Not sure it wouldn't just set things off earlier.
One thing that makes this one tricky is that it depends so much on public opinion, and on the opinions of people within the army and the rest of the government. How would they have reacted to any of the situations we've touched on? I don't know for sure, and I've seen no evidence that anyone else does.