Percy's amnesia in SoN gave him the perfect excuse to build his confidence to where it should be and break up with Annabeth, with very few consequences, and I will always stand by that. Annabeth may be easier for the Greeks to back due to her history there, but she has nothing at Camp Jupiter, where he quickly and firmly established himself as a powerful, capable and trustworthy entity, getting promoted to Praetor within a week. I honestly believe that, after his SON quest and the "greeting" Annabeth gave him on their reunion, Percy should've broken up with Annabeth, stayed with Camp Jupiter and thrived there.
Not only would the structure help him long-term in the way it canonically did in the (maybe) three days we saw him spend there, but he has a proper support system without biases in favour of Annabeth. It would also provide a fun narrative contrast to Jason, which RR was trying to push in-story, where they thrive in the other's camp despite the shadow of a figure they kind of feel they have to live up to. Jason does better in the looser structure of CHB due to his having picked up many wolf-ish traits from Lupa, while Percy almost *relaxes* in the strict routine of CJ because it helps with his ADHD symptoms.
And this confidence he very clearly has in SoN, and the way he almost romanticized the possible relationship between him and Annabeth when he didn't have his memories, would lead to a very jarring experience when his memories are settled and he reunites with Annabeth. Because, woah, that was not just a subconscious bias from preferring his new friendships/camp structure, but she actually is like that.
Boom, he now has recontextualised memories and perspective of Annabeth, and a place where she doesn't have a stronger reputation than him, and he can fairly safely break up with Annabeth and escape attempts at retaliation.
He might wait until after the quest the Seven go on, so that he can immediately cut her out of his lfe, but this new perspective changes how he approaches her.
The canon would never, but fortunately for us, there are fic writers. If I get enough time somewhere in this month or next, I might end up writing this, but I will have to see, no promises.
Back to your point; you have covered mostly everything so I don't see what I should be adding to this exact scenario but yes it is one of the few perfect pit stops for a percabeth break up to happen. I am going to go down a slightly different avenue for this one as my thoughts vary a bit from yours, so bear with me. It is going to be long, but I need to recap and clarify a few things for others; I will get to your specific questions near the end. [Everything's numbered, so if you are in a hurry, just scroll to the final four points]
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The breakup could have been mutual, too, if Rick had written it well. If we follow the proper timeline, Annabeth was still hung up over Luke up until the end of BoTL and had just started to get over him at the start of Last Olympian. Percy had just been informed of the Prophecy and under massive stress between end of BoTL and start of TLO and was obviously arguing with Annabeth over it as she repeatedly got angry at him for reasonably freaking out over his then "assumed death" prophecy. To find some relief from all this, he was dating Rachel, who he really liked, had a kiss with, and overall a good relationship with.
Then the whole plot of the Last Olympian occurs in what can be considered to be a few months at best. Then both Percy and Annabeth get together but to recap the events leading up to this:
Rachel breaks up with Percy despite their good relationship due to understanding the role she needs to play in saving the Oracle.
Luke who had previously visited Annabeth to make her join him and who Annabeth was somewhat hung over still, dies.
The war ended up killing a lot of close friends and companions and was a generally traumatic event for everyone involved.
Percy is very high on mixed emotions, reeling from everything that's happened when he gets together with Annabeth, so neither of them have had any time to process anything that's happened properly enough.
They have their one month anniversary on September 18th[The Staff of Hermes short story], and Percy disappears in October. [Riordan.wiki has years and dates of important events].
So they have barely been together for two months, and Percy's already missing.
This is Annabeth's thought process when they do reunite. It implies she has been doing a bit of thinking about their relationship and might not be as sure about it anymore.
She also seems to heavily dislike Percy's initial idea of settling in New Rome and seems to have no intention of taking a break from quests like Percy does.
She also seems to dismiss Percy's unwillingness to go to NRU after the news of Estelle's birth and the events of CotG.
Those were a few key points of contrast between them both on their future plans. Not small ones either.
Now, back to their reunion and the events following; here are a few things:
She judo flips him in front of the Argo II and almost entirety of Camp Jupiter.
If Annabeth didn't know that Percy had lost the Curse of Achilles then she was judo flipping him on the small of his back where his mortal tether was located ; which she knew about and therefore would have killed him in her anger and hyper emotional state.
If Annabeth did know that he had lost the Curse of Achilles then she was still judo flipping her boyfriend who had been missing for six months after surviving an already very traumatic event where he barely managed to survive and was then put through kidnapping, six months of amnesia, and brainwashing against his will by a goddess and then thrust into an unknown land with enemies on his trail.
Further, she blames Percy for disappearing for all of the things HE suffered through against his will and HAS THE AUDACITY TO MAKE HIM APOLOGIZE FOR IT.
Annabeth says and I quote that she likes keeping Percy on his toes.
She also agrees with Piper that Percy needs to be leashed and controlled.
Then Percy's side gives us a few more concerning tidbits:
Percy repeatedly notes that he is scared of Annabeth or thinks she might hit him
He also says that she often brings up Rachel to make him uncomfortable.
Percy also doesn't feel comfortable confiding in Annabeth about Gabe or past trauma.
He repeatedly feels he isn't good enough for Annabeth, and obviously Annabeth's behavior consciously or subconsciously on her part enables his thought process.
He ends up almost killing himself because Annabeth makes him promise not to use his specific ability despite it being used in self-defense by Percy, which ended up saving them both because she was afraid of Percy's powers.
Yet another thing to point out is that Annabeth's fatal flaw hubris and her abandonment issues feed too much into her behavior, and until that is fixed, nothing can be helped.
Annabeth punched Percy in ttc just because he didn't ask her to dance like she wanted him to. She also gave Percy a very hard time just because he knew another girl (Rachel) and immediately acted controlling and toxically possesive towards Percy over Rachel and Calypso despite her and Percy not being in a relationship at any of those points. In fact, despite her supposed crush on Percy, she defended Luke all the way till Botl, despite his repeated attempts at murdering Percy, and even went as far as to say Percy was unreasonably angry at Luke. She also displayed rude and downright awful behavior when interacting with Rachel; and Rachel, being the better person, handled her with class and grace. This all is not even including her demeaning remarks against Percy's intelligence and condescending behavior.
We can conclude from all this that Percy and Annabeth are not compatible, with different future goals and clashing perspectives and most importantly due to lack of proper communication, terrible misunderstandings, and Annabeth's consistent toxic behavior.
But neither the characters nor the author and not even most of the fandom acknowledge any of these glaring issues, so nothing can be done, but their break up can go several ways:
1.
Percy and Annabeth both break up amicably with Annabeth apologizing and understanding the gravity of her behavior and her mistakes and consistently making up to Percy for all the things she has been doing terribly. This is only possible for pre Tartarus or pre MoA situation.
Annabeth could have had a good arc in overcoming all of her problems while Percy too dealt with his self esteem issues and past trauma and they could have given their relationship a shot again in the future; they are both too young right now.
2.
Percy confronts Annabeth on her behavior and breaks up with her. Things get ugly, and it ends up being horrible for both of them. This is a traumatic situation on both ends, but the reason why this is unlikely is that Annabeth and Percy have a case of trauma bonding, and there's hints of codependency more on Annabeth's end. Which makes Percy mask Annabeth's toxic traits easily and for Annabeth to ignore Percy's devolving mental health and self-esteem issues that she has been enabling unknowingly.
3.
The other likely option is third-party intervention. Sally or Poseidon/Athena or other campers.
I also disagree that the camp half blood would back Annabeth. Percy's their [CHB] unofficial leader and a good friend to all of them plus he is the reason the children of other non-Olympian gods have cabins and get claimed early so he definitely has a higher status than Annabeth does in both camps. Camp Jupiter also does respect Percy more than any other Greek, and he is the only Greek to ever be made Praetor. So in social or reputation terms, Percy's winning.
But I don't think other campers would dare meddle in Percy and Annabeth's business. Maybe the seven would. [Leo and Frank would never. Jason probably doesn't understand the whole situation. Piper’s misguided and would probably feed into Annabeth's behavior, so either Hazel or Reyna? Hazel would definitely help Percy understand, and Percy would totally hear Hazel out. Thalia, hmm, I don't really know what Thalia would do, but I don't think she would blindly back Annabeth. She does seem to care for and respect Percy just as much. Nico isn't touching it with a 20-foot pole, plus I don't think he is over his Percy crush at all, so he's not an unbiased party either. ]
4.
Last option and what probably is the most likely is that things continue as they are and Percy eventually snaps due to endless pressure and his mental health issues and Annabeth can't deal with it and then all their other issues which they have both ignored come spilling out and then it's just a clean break but it's going to be the most excruciating way out and given their luck definitely happening.