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It's a pretty standard flaw for a child of Mars / Ares. We see something similar in Clarisse.
The wiki claims that Frank’s fatal flaw is his low self-esteem; but I would argue based on how Frank reacts to Octavian it is Wrath (much in the same way Octavian's is wrath).
I'll give you some examples of what I'm referring to:
Blood roared in Frank’s ears. He wanted to strangle Octavian with his bare hands, but at that moment, Gwen gasped. - page 142, Son of Neptune
Frank at this point has not spoken to Octavian on "screen". He doesn't know about Octavian's blackmail of Hazel. He doesn't even necessarily know she hates him. He just decides Octavian needs to die - as a way of avenging Gwen. It's a quickly made judgement and one with potentially significant consequences if Gwen hadn't woken up. Now, do I think he would have done it then and there? No. But he definitely considered it.
Frank snarled. “You backstabbing, slimy-” Both Percy and Hazel had to restrain him. - page 177, Son of Neptune
Once more, it's Frank who shoots straight to violence. Frank is threatening Octavian's life long before Octavian threatens the lives of the Greeks. It takes both Percy & Hazel to hold him back from potentially murdering or seriously injuring Octavian right there.
He worried about his friends in the Fifth Cohort and how they must be feeling as Octavian ordered them to march on Camp Half-Blood. Frank wanted to be back there, if only to stuff a teddy bear down the throat of that slimeball augur.
- page 483, House of Hades
Mark of Athena has no POV for Frank, so we don't hear his internal thoughts until House of Hades. Frank doesn't actually think about Octavian a lot - that is, until then end, who he considers killing the guy three times in less than 100 pages.
Leo clapped Frank on the shoulder. “Way to go, Zhang. Now you can order Octavian to fall on his sword.”
“Tempting,” Frank agreed. - page 565, House of Hades
So again. Interestingly Leo even knows that Frank has the desire to kill Octavian. He plays it as a joke, but Frank doesn't. He says "Tempting" which isn't a commital. But then his own internal narrative says, "agreed". Like, "yes, I'm definitely going to abuse my new found power to kill this guy"
“When we take over command,” Frank suggested, “our first order of business should be to load Octavian into the nearest catapult and fire him as far away as possible.”
“Agreed,” Reyna said. - 572, House of Hades
These are the ones I found while flipping through. Obviously Frank doesn't have a POV in Blood of Olympus so it's not as likely to come up.
However, the amount of times this is brought up makes me think that it’s possible Frank would kill Octavian if Octavian had survived.
Maybe that’s why Octavian had to sacrifice himself, because can you imagine if Octavian was murdered by a newly returned hero?
The fallout of that would be huge - New Rome would have destroyed itself. Which is the opposite of Octavian’s goals.
Just a thing to consider.
But my main point is that I don't think Frank has run into issues with his true Flaw just yet. The problem resolved itself before he needed to do anything about it. But that doesn't mean he never accepted that Octavian was a hero.
Help me out here, if you made it this far, what is the situation in Camp Jupiter? Are their tensions? Is Octavian a hero to them - or still a villain?