This is a really good question! Thank you so much! There are a number of things to think about here. Before I get started, I have to admit I can't directly answer your question with certainty as Caesar never wrote about Cleopatra.

With that said, I think I can answer some smaller but relevant questions fairly well. I hope that this helps answer some part of your question indirectly!

1. Was incest considered morally wrong in Ancient Rome? Not only did the Romans think it was wrong, they thought it was WRONG in all caps. They were very against it. Roman law considered any sexual activity between parents and children or among siblings to be nefas, which is to say "unholy" but in a legal way. This rule applied equally to biological relatives and relatives by adoption. Roman law also forbid formal marriages within certain degrees of consanguinity. I'm not an expert on Roman law and the language is very specialized so I can't tell exactly how how distant cousins had to be before they could be married, but it seems like marrying first cousins or nieces/nephews is right out. To this point, rumors of incest could be very dangerous to one's reputation and career. The emperor Domitian, for instance, was plagued by rumors that he was carrying on a secret relationship with his niece. He went to great lengths to put these rumors down, but they kept coming up.
2. Why did Caesar have a relationship with her if he thought incest was wrong? It is really, really important to know that it is far from certain that Caesar even had a sexual relationship with Cleopatra and, if he did, how long it lasted. Yeah, Cleopatra claimed to have had Caesar's son, but she was known to be dishonest and vicious (vis. what she did to her brother/husband). The only proof she could provide of Caesar's paternity was her claim that Caesarion resembled him, which is obviously questionable. While he never mentions it directly, we have some evidence that when he was still alive Caesar considered Cleopatra's claim to be nothing more than a rumor. Rumors about Caesar's sexual promiscuity were a dime a dozen, so the mere fact that the rumor existed doesn't really say much (scroll down to 50-52 in the link). On top of that, several of Caesar's friends wrote and distributed pieces claiming to provide evidence that the child wasn't in fact Caesar's. So, in the end, all we have are a lot of contradictory claims and a dearth of reliable sources. There is a clear motive for Cleopatra to claim she had Caesar's son, and there are clear reasons Caesar would deny paternity. Different Classicists give different opinions and I'm not an expert in this specific area so I won't weigh in.
3. What did Octavian think? Well he needed her out of the way anyway. I don't think the fact that he had Caesarion knocked off lends any kind of credibility to Cleopatra's claims, though. Augustus was just a guy who didn't leave loose ends.
4. What was Antony thinking? I can't give you a good answer here. We can't really trust anything Roman historians have to say about Antony, or the relationship between Antony and Cleopatra. As you probably know, their "marriage" created a huge scandal on the Roman side as it looked like Antony was abandoning Rome and it left Antony's well-connected, intelligent, and very faithful Roman wife Fulvia in a horrible position. Consequently, there aren't really any contemporary historians who we can trust to give us an accurate description of their relationship. They all hated Cleopatra and they hated Antony even more.

So, in the end, I can't tell you why or how Caesar and Antony rationalized their relationship with Cleopatra, or if Caesar even had a relationship with Cleopatra. But I think Horace's very famous poem helps wrap things up neatly. In the eyes of Roman elites, Cleopatra was the ultimate "Oriental." To someone like Horace, that would mean she was sex-obsessed, selfish, and just plain wrong. I don't think it's a stretch to say that Horace was taking a swipe at Eastern marriage practices when he talks about her "flock of men, hideous with disease."