Then of course it is my partners turn to do the ikkyo- I could also use my strength and knowledge of the techniques and block and walk (I did it once a few weeks ago just to try show him and he could not move me either) but I of course do not- I really want to avoid these ego games. But the end result is that I look like the complete loser of the dojo.
For the most part, everyone else in the dojo is too busy worrying about themselves to care what your technique looks like. On the other hand, the best way to get branded as a "loser" -- short of injuring people -- is to do as your partner was doing.
Learning to deal with frustration is part of the practice. In my experience, "escalating" by being just as stubborn as your partner rarely works and can often lead to ramped up aggression on both sides, and therefore a higher risk of injury.
As for this specific incident... of course the instructor is more competent than you are. That's why he's the guy standing in front of the class and you're not. While it's possible that uke decided to be less stubborn simply because he was dealing with the instructor, it's also likely that there are some specific differences between the instructor's technique and yours that make resistance less possible/effective. Your job is to figure out what those differences are.