"Report of Capt. G. V. Fox, U.S. agent, of second expedition for the relief of Fort , Sumter.

STEAMER BALTIC,
New York, April 19, 1861.

…We arrived off Charleston the 12th instant, at 3 a.m., and found only the Harriet Lane. Weather during the whole time a gale. At 7 a.m. the Pawnee arrived, and, according to his orders, Captain Bowan anchored twelve miles east of the light, to await the arrival of the Powhatan. I stood in with the Baltic to execute my orders by offering, in the first place, to carry provisions to Fort Sumter. Nearing the bar it was observed that war had commenced, and, therefore, the peaceful offer of provisions was void…"

O.R. Series I, Vol. I, Chapt. 1, p. 11

Assuming Lincoln was going by Fox's report, it is easy to see how he could regard the expedition as not having arrived before Beauregard commenced the bombardment. Note that the Baltic stood in after the Pawnee arrived at 7 a.m., after the bombardment had begun. From the report, it sounds as though little was visible to ship at sea the night of April 12 and it was not evident that the attack had begun until the Baltic approached the bar sometime after 7:00 a.m. According to Capt. Foster of the Engineers, the first sighting of two ships off the bar was 1:00 p.m. on April 12.

Of course the other possibility that we haven't discussed is that by "arrived" Lincoln meant "at Fort Sumter"...