Including Non-Jews in the Bris Ceremony?

Being one who only accepts the view of matrilineal descent determining Jewishness, I follow the view that when the baby's mother is not Jewish, the baby is not Jewish and is not required to have a bris. 

However, it happens often enough that the mother IS Jewish, and the father is not a Jew. Which means that the baby needs a bris, and I am brought into the situation.

In the email I send to parents pre-bris, I include the line that "All participants in the ceremony should be Jewish." That instruction is both for those who want to include their non-Jewish friends in the ceremony, or their non-Jewish relatives.

How could I be so exclusionary?

The way that I put it is this: In the hypothetical case where I am invited to a baptism (it has never happened and probably will never happen, not that I would go), I would be happy to watch a ceremony/service of a different faith, one which I do not understand, nor wish to participate in. Watch? Sure. But I would feel wholly out of place participating in the ritual.

Now, I understand that people who are not Jewish often love to participate in Jewish rituals, whether it's attending a Passover Seder or a public event of lighting Chanukah candles. In these cases, however, the roles are exactly as I described. As a bystander, watching, albeit close by.

The same thing applies at a bris, with one exception. Non-Jews are welcome at a bris. They can watch, they can listen, they can ask questions. But the ritualistic components of the bris are for Jews. Even the baby's father, who plays a significant role when he is Jewish (which is the case, thank God, most of the time), does not have any requirement or real role in the ceremony when he is not Jewish!

HE is not commanded, HE does not say blessings, he can't appoint the mohel to be his agent to fulfill a mitzvah he does not have. 

So here it is:

Non Jews are welcome to be present. 

Non-Jewish relatives should be mindful that the ceremony takes 5 minutes (or so). They'll have a lifetime to be in this child's life and to take all the pictures they want. 

Non-Jewish friends - in most cases they are more than happy to be on the sidelines, even if baby's parents feel very close to them. They need not actively participate in the bris. 

As for the baby's non-Jewish father, I do not deny a father the opportunity to hold his baby. As soon as the bris is over, unless the baby's mother wants to hold her son immediately, I give the baby to his father to hold (assuming of course that he is supportive and present - in some cases the father is in reluctant agreement and is not present).