Okay here are my survival chance ratings from lowest to highest.
Kacho & Fuugetsu: Melody is their only chance of escape. Keeni will protect them, but he is already worn out and is not allowed to help them escape. There's one other Pro Hunter in Kacho's guards, but aside from him all others are useless when it comes to Nen defense. One of them might survive for now, but the chances that both of them do is very low imo. Especially since we had lots of deaths already but not a real shocking death since Shalnark and Kortopi. And shocking deaths are somewhat iconic for HxH.
Tyson: Is too freeminded and doesn't take the succession war serious at all. Tyson's nen beast drains aura from her guards in exchange for happines, that's such a shitty ability even Izunavi can't protect her if his aura was drained. I would give her the highest death chance, but she was barely shown at all and I don't think any prince will die without his time in the spotlight.
Zhang Lei & Tubeppa: Both take the succesion war very serious, but they are too cautious and not ruthless enough. They have no defense against Nen at all, but they do have professional guards loyal to them. Zhang Lei's ability seems to be too complicated to be figured out by a non-nenuser like himself, and from the little description that we have of it, Tubeppa's beast falls into this category, too.
Luzurus: Seems to be much fitter than his siblings think him to be. Was shown the least of all princes, which is the only reason I give him a higher ranking than his siblings above.
Camilla: If she wouldn't be such a powerful nen-user, she would be dead already. However it is inevitable that her own mindset will lead to her demise. Just not so soon.
Marayam: Has an ideal ability for protection and has Biscuit (!!!) and Hanzo as his guards. Biscuit has the captain of his guards by the balls, which means she can organize his defense, and since she probably is the best nen user on the deck, we can expect a very good defense strategy.
Benjamin, Tserriednich, Halkenburg, Woble: I believe those four princes are relevant for the endgame and as such will die last.
Bold prediction:
Which prince will win? I think Benjamin or Halkenburg, depending on wether Tserriednich succeeds in killing Halkenburg or not (he will definitely try). Woble will survive because the last coffin of that weird machine will be filled by Morena.