The Inheritance Feud

Rashi says that when Sarah heard Yishmoel laying claims to a double portion of yerusha because he was the bechor she said to kick him out.  Rashi says that Yishmoel was doing averot chamurot.  However, Sarah specifically notes that he will not inherit with my son, indicating it was the inheritance issue that drove Yishmoel out.  Why is this the big issue, more than the averot, which cause Yishmoel to be kicked out?  

Rav Elchonon (in volume 2 of Kovetz Shiurim) say that when it comes to yerusha of a bechor there are two aspects.  One aspect is that he inherits because he is related.  The other aspect is that the son stands in the place of the farther, as the Gemorah says קם תחת אביו.  When it comes to a gentile, there is a biological relationship, but there is no concept of קם תחת אביו and hence there is no halacha of a bechor getting two portions.  When Yishmoel was claiming he was the bechor, it was not a mere claim on the wealth of Avraham, it was a claim to the legacy of Avraham.  He was claiming I stand to take over Avraham's life mission, I will carry on his legacy.  That lack of recognition of his place and trying to usurp Yitzchak's right as the true son of Avraham to carry on his legacy is what prompted Sarah to demand his expulsion.  This is why Yaakov felt compelled to ensure that he was the bechor and not Esav for it is the bechor that carries on the father's mission.