In this article published right after that of Rabbi Ralbag, Rav Yaakov Epstein comes out against the lenient ruling regarding children born to single women who have been artificially inseminated.
Rav Epstein approaches this issue mainly through the perspective of the issue of שתוקי, and he main conclusion from his analysis of this issue is that a שתוקי is generally dealt with strictly, and is considered to be at best a ספק ממזר, and that a child whose father is not known due to AI almost definitely falls into this category.
With regard to the notion that a woman who goes to a hospital to be artificially inseminated is dealt with as if the case is one of כל דפריש מרובא פריש, Rav Epstein raises a doubt as to whether this is really the case. He notes that in order to invoke פריש we rely on the idea that the semen is no longer connected to its donor. However, since the woman does come to an established location (the hospital), there may be room to say that she is really entering into a case of כל קבוע כמחצה על מחצה דמי. Even if this is not a true case of קבוע, and even if it thus falls into the פריש realm, since we are dealing with a case of ספק ממזר we are going to be strict.
Another line of leniency would be to allow the doctor to testify that the donor is someone who would not render the child a ממזר. However, Rav Epstein notes that this is nearly impossible for a number of reasons. First, sperm banks rely on the anonymity of the donors, and thus the doctors are unlikely to be able to make such a statement (and they would need to know not only the identity of the donor, but also be able to confirm that the semen implanted in the woman definitely came from that donor and that the woman had not received any other semen). Furthermore, Rav Shlomo Zalman Auerbach ruled in such a case that there are at least two problems that would remain - the suspicion of זנות on the part of the woman [ed. - would this change if such procedures become more common?] and the fact that since the semen is obtained via הוצאת זרע לבטלה that this would somehow convey a bad influence on the child that would be born as a result. Finally, Rav Epstein notes that given the various halachic implications of fatherhood, there is a necessity for a fairly high level of נאמנות in establishing paternity, and such a level is virtually impossible in this case.
Finally, Rav Epstein notes the position of Rav Moshe Feinstein which claims that the status of ממזר can only be conferred via intercourse and not through AI. He rejects this position, noting that most other poskim do not follow this view.