The thing about percy is that yes he's suicidal but he also genuinely wants to live.
His desire to die is opportunistic; it manifests itself in life or death situations — in the first saga, usually when he's alone, so his death won't cause the immediate death of someone else — like in the first book, when he jumps from the arch to try to save the family that was with him, and when he falls into the water, he says he doesn't know why Poseidon had saved him and that he feels like drowning himself; he also wishes he was able to drown when clarisse's boat explodes in SoM, after Tyson dies.
This changed in the second saga, when we see these moments through the eyes of other characters (Percy almost letting himself drown in Tartarus, Percy letting himself be poisoned by Kym and (one could argue) Percy lowering riptide when confronted with Tartarus — tho Percy letting himself be poisoned is the main one, bc he was being influenced by magic during the other ones.)
Also, there's a pattern there – his suicidal tendencies are passive, usually it's him wishing something had been able to kill him or considering stop fighting for his life and letting something end him. It can also manifest itself through escapism – in the first book, bf thinking about drowning, he thinks "maybe I should
just stay down here with the catfish, join the bottom feeders", and in the tbotl, he considers staying in Ogigia and letting everyone think he's dead – these thoughts are also a form of suicidal ideation, and they also fit the pattern of being passive – he thinks of staying in some place when he's already there, but doesn't think of actively running away and disappearing when he's at camp or at home.
There has definitely been an escalation in how far Percy lets himself be led by these thoughts – he actively almost gets himself killed in BoO, and would've died if not for Jason. Still, my point about him wanting to live is an important aspect of his characterization – he thanks Jason for saving him, he's not mad and, apparently (tho we can't be certain, bc it's not his pov), is not disappointed at being saved. He wants to be saved. It's just that sometimes, he doesn't feel like saving himself, he can't convince himself to.
Another important pattern is that his suicidal tendencies are closely linked to his feelings of guilty, not insatisfaction with his life — in TLT, he thinks he should drown bc he likely wasn't able to save the family in the arch, in SoM, he wishes he could drown after Tyson's death bc he also feels guilt over it and the way he treated him before, in BoO, he lets himself be poisoned bc he thinks he deserves to die like this bc of what he did to Akhlys, and he considers staying in Ogigia bc (amongst other reasons) he's too dangerous, as shown by him blowing up mount saint helen. Which is why I say he's suicidal, but he wants to live — the moments he considers death are related to him thinking he doesn't deserve to live, instead of not wanting to. "You didn't do X thing or you did do Y thing so you don't get to have this privilege (living)".
So, basically, he's not the type to look at minor (or major) inconveniences and think "I should kill myself", he's the type to look at mistakes he made and think "I deserve to die and something probably should kill me right about now".
