Stark vs Lannister, military challenge, including Robb (Core). Stark wins

Lannister plays "A Lannister pays his debts", and kneels a character to kill Robb

Can Robb still use his ability to kill a Lannister character before he leaves play?

Yes. Robb becomes Moribund but he's still on the table until both of you pass consecutively and the window closes.