Something besides dying had to eliminate Kate as a candidate. Sawyer said he saw her name but it was crossed off. Widmore knew that Kate was off the list as well. The reason Kate is off the list is that a female candidate cannot have a child.
Kwon is a candidate but that could mean Jin, Sun, or Ji Yeon. MIB thinks Sun could still have been a candidate when he spoke to Sawyer in his cave because he did not know she gave birth because it happened off the island. We never see Claire’s name as a candidate because she gave birth on the island. MIB & Jacob’s “adoptive” mother could not have her own children according to this rule which is why she stole the twins from Claudia to raise new candidates.
Widmore would have witnessed off island the return of Kate with her child Aaron in the Oceanic 6 story. Thinking she gave birth to him like the cover up story she told, he’d dismiss her as a returning candidate.
When she returns to the island she’s automatically eliminated as a candidate. Why? Because Jacob & MIB know she is pregnant with Jack’s child (they spend the night together in episode 316, right before getting on Ajira 316). Kate will leave the island to give birth to Jack’s son and name him David.