It's gonna be incredibly awkward for ZAFT when Athrun's teammates, Rau and Patrick Zala hear news of him defecting if Cagalli awakened him.
I was thinking that the best opportunity, if she can't find a way to privately do it during this fight (i.e. grappling in the cockpit of the Aegis) would actually be when he's sent back in to Heliopolis to destroy the Strike.
Cagalli would need to fight Athrun, while Kira, Mu and Sahaku focus on stopping the rest of the GINN squad from blowing up Heliopolis. If she awakens Athrun while soloing him, she can plausibly disable the Aegis (or pretend to disable it) without killing him and then claim that she beat him the old-fashioned way and then take him hostage as a prisoner of Orb (not the Earth Alliance).
Being able to present him as a prisoner would be the ideal scenario. Athrun Zala, being the son of ZAFT's Defense Chairman, is just as high-value a target as Lacus Clyne. Cagalli could probably be able to use him to get Rau to back off, offering to give him back if Le Creuset retreats from the colony immediately with the 3 other suits he already has. As probably the highest-ranked Orb official there apart from Sahaku she would have to be taken seriously.
From there, we have a lot of advantage.
If Athrun goes back, we now have a trojan horse in ZAFT. If Rau is insane enough to refuse the exchage and try to sacrifice Athrun, we now have Alex Dino who has a plausible motive for betraying ZAFT - they betrayed him first and left him for dead.
Personally, I don't think Rau could even give that refusal without Athrun's teammates having extreme objections to the point of some of them mutinying over it. Athrun would have to cut in front of Yzak to get a piece of Rau.