Lincoln was very much aware that sending a supply ship to Fort Sumter would escalate the tension between the North and South.
Why did the South attack Fort Sumter?
Great efforts were made to avoid an armed conflict between the North and South. Politicians from both sides exerted efforts to resolve the conflicts between the North & the South. The problems were enormous and included :
A. Slavery;
B. Economic competition ; and
C. The divisive tariff issue.
South Carolina, in their view, had endured an affront to their sovereignty by "foreign" forces, meaning the Union army, on their soil since December 20, 1860 when the state seceded from the United States and then joined the Confederate States of America.
The Charleston harbor was an important one. In order to secure this Union fort for the South, the fort was bombarded with canon & mortar fire until it finally surrendered.
In summary, Fort Sumter had to be taken over by the South as one of many steps required to establish a new nation, the Confederacy.