The cabinet opposed resupply at first. When they voted again almost two weeks later, some changed their mind and a narrow majority supported Lincoln.
Some people on the Internet have major axes to grind and you can't always trust what they say.
They didn't change their mind about it causing a war. They changed their mind about whether it would be in best interests of the Washington DC Government to start a war or not.
Their boss wanted a war. So what did you expect them to say once he made his feelings plain and they served at his pleasure.
They still thought it was going to cause a war. None of them thought 8 armed ships would deliver "supplies" without triggering that war.
Major Anderson himself said it would cause a war, and he certainly indicated that he believed the Union started it.