||My family's criteria is that a potential shidduch must be a Jew according to the standards of the Israeli Rabbinut.
If the conversion is recognized by the Israeli Rabbinute and the person is eligible for marriage to a Jew in Israel, then of course WE and NO ONE else would have any right to reject that person. Chas V'shalom. Our standards are CERTAINLY NOT more stringent than those of the Rabbinute of Israel.
Of the two dozen or so children of converted women who have been presented to us by various shadchanim, NOT A SINGLE ONE of the conversions was done by a Dayyan on the "Accepted" list that is published by the Israeli Rabbinut. (the complete list is online at ITIM.il).
Ceremonies of conversion that were performed by Rabbis or organizations who are not on this list are not recognized in Israel. Our family has also verified the authenticity and completeness of this list firsthand with the secretary at the offices of the Rabbinut in Jerusalem.
Shall we make shidduchim for our sons that would result in children who would be NOT JEWISH according to the Israeli Rabbinut?
Shall we marry our daughters to men who are GENTILES according to the Israeli Rabbinut?
Our family's Rav has also informed us that a Kohen is not permitted to marry a girl with a Gentile father and that he will not perform a marriage between a Bat Kohen and a man with a Gentile father. (Mr. Confused is a Kohen). Are we incorrect to follow this Rabbi's directives?
I did not consider the case of a boy or girl who has a Gentile maternal or paternal grandfather which does not present a halachic problem for any Jew.
This particular set of circumstances has not been presented to us is because only 8% of American Jews who are the product of an intermarriage (only a Jewish mother and Gentile father can produce a Jew) affiliate as Jews and marry another Jew. Statistically it is pretty improbable that such a young person will be in the shiddach market in any observant community.