Nah, he still got a chance. He has to flip it again with something like "Thank you for the opportunity sir, but I still have a duty I must fulfill for you, no the young master! I can't allow myself to escape, not until the young master wakes up!"
Yeah just say "I can't kill myself until I find Kaiji", also flip it back on hyoudou too and say killing himself would be failing Kazuya. Actually say you can't die instead, so hyoudou doesn't just go "well if you can't kill yourself, I'll just do it for you". And say if he were to die he would be failing both Hyoudou AND Kazuya.