"I will not lose to you" probably more of Vegeta then Goku, seeing as how this guy came out of nowhere and Vegeta has a habit of getting beaten by people that just pop up. Androids, Buu, Baby, Super 17. Its even more true in the movies, like all except Meta Cooler and Super 13. Im sure his pride is screaming at him to beat a newcomer for once, especially one claiming to be his older brother and a fellow saiyan survivor.
I wanted Gogeta's personality to be exactly what he is: a composite of Goku and Vegeta's personalities. Therefore, by that logic, at times aspects of one or the other's personalities would become more dominant than normal