Hmm… No, it’s a good question. I don’t consider post-LR Hope to be suicidal, not after he’s at last with Lightning by his side, but I do consider post-XIII-2!Hope and LR!Hope to be like that. I’ll explain myself.
About post-XIII-2!Hope, aka in the Chaotic Age, I can see Hope when falling in his deep depression that later allowed the rose-haired phantom episode harboring such thoughts. Think about it: he’s burdened himself with the fate of all humanity and the world, and no matter how much he does, and how much he sacrifices, it doesn’t work at all. Plus, each one of his friends didn’t notice at all that Hope had fallen into depression, which means he was completely alone. I can see Hope thinking of suicide to escape from all it. Until the phantom hooked him, though.
And LR!Hope, it might be subtle, but it’s actually in-game. Though he’s resigned about it, he’s well aware he won’t live see the new world. And he’s… pretty willing to go through that. When he bids farewell to Lightning, he insists her “it’s too late for me now” and “please, forget about me and save all the rest”. He’s not asking for salvation: he’s asking Lightning to let him die. Well of course his death would’ve been needed if Light wanted to kill Bhunivelze -even though she managed to take a third option- and Hope’s both heart and logical mind would tell him it was the best: his logic mindset of “better save hundreds than save only one” and his heart by “Light must be free and I’m an obstacle”. But, who knows. If things went as he set, Vanille would’ve performed the Soulsong and all souls would’ve been saved except his, trapped inside Bhunivelze. And Hope’s plan was Lightning to slay God without taking the risk of saving him during their showdown. If she’d done as Hope had in mind, then his soul would’ve been destroyed with the old world. And that’d mean not only his own complete obliteration and cessation of existence, but that everyone, every last being in the universe, would have forgotten he ever existed once. Perhaps, after all he went through, Hope thought he’d had enough of misery and it was time to finally forget about everything and let others forget and free everyone from his shadow, including himself.