I know what you say makes perfect sense to you, but this isn't the way the real world works. They are already a family unit; YOU are the one who wants "in", which means YOU are the one who is going to have to suck up a little. You will get your chance for unconditional respect when YOU have a daughter who is dating someone that wants into the family you have created. Right now, it is her father's turn to be the head of the family.
Don't be a typical young male who is intent on proving he has the bigger penis. In the end, you will lose, my friend. Blood is thicker than water (unless you are talking about a seriously dysfunctional family). Your girlfriend may say she is right next to you for now, but this will NOT be the case once she starts having babies. Trust me on this. She will resent that there is strife.