A gut gebnsht yor and a gut mo'ed. Thank you for your post today describing the problem faced by kiruv organizations. You indicated that you would like to see source material on this problem, so here is my survey of the relevant halakhic literature:
R. Moshe Sternbuch, Teshuvot ve-Hanhagot, V, no. 296 rejects the conversion of the child of an intermarried family unless the mother converts together with the child [-including that the mother sincerely accepts all the mitzvot as being inherent to the conversion]. Rabbi Sternbuch explains that by the Beth Din attempting to convert the child without the mother, the Beth Din is essentially telling the Jewish [so-called] father to continue to commit the yehareg ve-al ya'avor transgression of remaining in the same home as the Noahide mother. Hence, the Torah will not recognize such a conversion, and the conversion attempt fails.
In Ha-Pardes Vol. 61, no. 10 (Tammuz 5747), the Agudat Rabbanim is reported to reject the conversion of a child from an intermarried family unless the mother converts together with him [-including that the mother sincerely accepts all the mitzvot as being inherent to the conversion]. How can children be validly converted when the so-called Jewish father is obligated to commit suicide rather than remain in the home of the Noahide mother [as per Rema to Shulchan Arukh Even ha-Ezer 16:2]. Available online at
https://hebrewbooks.org/pdfpager.aspx?req=12413&st=&pgnum=32&hilite=
To better explain the words of Rabbi Sternbuch and Agudat ha-Rabbanim, it seems to me that the prooftext is as follows. The Gemara states lo amrah Torah shale'ach le-takalah in three different contexts: Yoma 67b regarding the scapegoat, Kiddushin 57b regarding tzipporei metzora, and Chullin 115a and 140a regarding shilu'ach ha-ken. Thus, by the same token, the Torah cannot recognize a conversion of a child of an intermarried family, because the worst takalah imaginable is that a Jewish gentleman thinks that he can establish a normal family with a Noahide lady, taking pride in the so-called "Jewish children" he has from this intermarriage. There is no way that the Torah would recognize such a conversion of the child [even according to Abaye (in Temurah 4b) that kol milta de-amar Rachamana lo ta'avid im avid mahanei. Abayei is discussing a one-time transgression, which he is willing to tolerate post facto, but not a permanent takalah of nasui nokhrit which is yehareg ve-al ya'avor. And how much more so should this be true according to Rava, who opposes Abayei there in Temurah].
In addition to teaching us that it is yehareg ve-al ya'avor for a Jewish gentleman to marry a Noahide lady, Rema (Even ha-Ezer 16:2) also writes that such intermarriage has a hefsed which is not found in the other arayot, since the resulting child is gentile. Yet, why must the Rema bemoan such a hefsed; let him simply recommend converting the child after he is born by immersing the child in the mikveh under the supervision of a Beth Din! Obviously, the answer must be that such a conversion will not take effect, and so Rema will not propose it as a "solution."
Based on points 1-4 above, it is clear that R. Joshua Hirschorn, Teshuvot Mi-Ma'ayanei Yeshu'ah no. 57, p. 161, three paragraphs beginning s.v. ve-hineh [available online at
https://hebrewbooks.org/pdfpager.aspx?req=1545&st=&pgnum=160 ]
and R. Yehudah Herzl Henkin, Teshuvot Bnei Vanim, II, no. 36, sec. 3 [available online at
https://hebrewbooks.org/pdfpager.aspx?req=20022&st=&pgnum=139
, are both completely wrong (be-mechilat Kevod Toratam) to claim that the child can be converted.
Thank you and chag same'ach,
Shalom C. Spira