Muse

First of all, there were 2 casualties in that battle, Liam and Soleil, yet only Liam is ever brought up.

Second, as a rider and as a squad leader, it is Dain's duty to investigate and report on any suspicion of breach of conduct or treason and Xaden and co going to Athebyne is a Codex violation. If he'd also known they were snuggling weapons out he would have been duty bound to report that too cause regardless of how noble the intentions are, it's still treason.

Third, if your bff is acting out of character and decides to start hanging out with a guy with the reputation of being ruthless and cruel, who used to want her dead, who has given you exactly 0 reasons to think he's actually trustworthy, you'd snoop around too to ensure said bffs safety. That's just basic.

Fourth, as the superior officer in charge of war games, Colonel Aetos bears full responsibility (legal and moral) for anything that happens during war games, especially anything that happens as a direct result of an order or negligence. He's responsible for Liam and Soleil dying.

Fifth, as the superior commanding officer in the battle of Resson, Xaden Riorson, wingleader of Fourth Wing, is responsible for anything that happens as a direct consequence of his orders or negligence. He brought two first years past the wards, chose to not send them away/tell them to stand down when it was clear there might be a real battle happening and allowed his whole riot to fight an enemy that outmatched them all. He's, therefore, at the very least partially responsible for the deaths of Liam and Soleil.

Sixth, obviously the venin are guilty of directly killing Liam and Soleil. I don't think that's something anyone would actually dispute but they're rarely mentioned in relation to Liam's death. But both Liam and Soleil are partially responsible for their own demise since they chose to fight knowing dying was a very likely option. Obviously no one is gonna blame them.

Seventh and Last, Dain had no authority to change, chose or organise anything related to War Games in general or the Battle of Resson in particular. He had no knowledge that his father would NOT follow the Codex for dealing with the reported violation and would choose to send cadets to their deaths instead. He had no say in who went to Resson (remember he did try to argue about Violet going and Xaden pulled rank), as that was 100% Xaden's choice. This means that he can't be held responsible for anything that happened at Resson.

The only thing Dain can actually really be held responsible for is violating Violet's privacy by reading her memories without consent. That's it. The rest was either his duty or completely out of his hands.