I misread the initial question. I was thinking that the situation was two separate games. I.E. the first game was Proximo, Leo, Dextarius and Kephros. And then the second game the player decided to simply remove Kephros, and still wanted to claim the Ludus benefits. I see now that the question is if Kephros dies in the course of the game, does the rest of the cohort lose the benefits. And no, I do not think they would. If Kephros started the game in the cohort but dies, the beasts would still get to use Will of Antony.
Well, see my example earlier where ludus benefits would be lost, even though the cohort still contains standing members of the correct ludus, just because their percentage dipped under 75%. It's not just a Morituri issue.
I don't think once a game has begun, that you can lose your benefits at all, regardless of who dies. Once a model is in the cohort, it is still in the cohort for model %, even it if subsequently dies in the course of the game. So in your example of 3 LM gladiators and 1 Legio gladiator in a cohort, that ratio never changes. Even if you lose 1 LM gladiator in the game, that model is still a member of the cohort.
The beast cohort would have to include a Morituri gladiator at the start of the game to be able to claim the benefits. I don't understand how a cohort with no ludus gladiators could claim any ludus benefits... a 4-card cohort of Proximo, Leo, and Dextarius certainly couldn't claim any.
Right, I agree. That 4 card cohort contains no Ludus gladiators and thus should not be able to claim any Ludus Benefits.