As I understandd the situation, Governor Pickens of South Carolina WAS suppling Fort Sumter with necessary food while peace delegations were in Washington. It was when those talks broke down that the supplies from Charleston stopped and the situation became critical inside the Fort.
I feel that the failure with both davis and Pickens in this situation is that while there was a Picket Boat at the Harbors mouth they had several months to equip more and to strenghten harbor defenses at the Harbor mouth such as booms and more heavily armed ships which were surely sitting in Charleston Harbor.
If a federal relief fleet would have had to have fired on a Picket boat and Confederate defenses at the Harbor mouth the south would not then have fired the first shot. The Confederates concentrated on removing the threat in the Harbor (Fort Sumter) instead of circumventing it, such as cutting off any possibility of relief from outside.