I have always read 1 Cor 9 as an extension of 1 Cor 8, which focuses on whether eating food sacrificed to idols is permitted. He says that it’s fine to do so if you feel it is fine; if you feel it is bad you shouldn’t. And if you feel it’s fine, but know someone who doesn’t, then you should be considerate about how you act around them so that you don’t entice them to do something they think is sinful. He describes the people who won’t eat meat sacrificed to idols as “the weak.”
In 1 Cor 9, then, I think it makes sense when he talks about becoming like the other people that he’s talking about their dietary and religious standards. When he’s with the Jews, he follows the Jewish law and eats kosher. When he’s with the Greeks, he eats like them. When he’s with “the weak” he follows their requirements.
I don’t see the cited passage to mean anything dishonest. In fact, it sounds like Miss Manners would agree - bringing ham for dinner with your Jewish friends or a bottle of wine to your alcoholic buddy are both really bad ideas, even if you’d consume both at home.
It’s also worth noting that in his epistle to the Philippians (which is generally agreed to have been written by Paul), he quotes what is apparently an even older creed or hymn in 2:5-11:
*He existed in the form of God, did not regard equality with God a thing to be grasped, but emptied Himself, taking the form of a bond-servant, and being made in the likeness of men. Being found in appearance as a man, He humbled Himself by becoming obedient to the point of death, even death on a cross. For this reason also, God highly exalted Him, and bestowed on Him the name which is above every name, so that at the name of Jesus every knee will bow, of those who are in heaven and on earth and under the earth, and that every tongue will confess that Jesus Christ is Lord, to the glory of God the Father.
So Paul seems to be affirming that Jesus “existed in the form of God,” whatever that might mean, and humbled himself by taking the form of a man.