I unhesitatingly rule that it is performed on Shabbat. The gemara (Keritot 7b) explains the pasuk from Parshat Tazria – כי תזריע וילדה זכר – עד שתלד ממקום שמזרעת, namely that a mother is only tamei’a (ritually impure) from tumat leida (impurity that is created by the birth process) after vaginal birth, but not if she gave birth in any other manner. Elsewhere, based on the juxtaposition of the laws of tumat leida and brit mila, the Gemara (Shabbat 135a) connects the obligation to circumcise on the 8th day with the woman being rendered temei’a. Thus, the gemara concludes that only a circumcision for a baby born through vaginal birth is done on Shabbat. These sources, however, only address the manner of birth, not conception, and there is no basis in the Gemara that when the conception takes place in a “non-natural” way that it should have any impact on the halakha that brit bi’zmano is doche Shabbat (conduction a brit on the 8th day overrides Shabbat).

There is an opinion brought in poskim that a child conceived through IUI should not be circumcised on Shabbat. This opinion is brought by Rav Shlomo Zalman Auerbach (Noam 5718 [1958]; Minchat Shlomo 3:98:4) based on a Rabbenu Chananel (Chagiga 16a) who states that a woman who became pregnant in a bathhouse is not rendered ritually impure due to the miraculous nature of the conception. Using this principle, Rav Shlomo Zalman Auerbach concludes that the Brit mila for any child conceived in any unconventional manner, such as IUI, would not override Shabbat.

I don’t think we have the power to make new derashot, or that we should worry about this obscure Rabbenu Chananel not cited by any of the Rishonim. The strong consensus of poskim is also that the Brit is done on Shabbat, and I have never heard that in practice people do otherwise (see this article where he cites Rav Ovadia Yosef, Rav Asher Weiss and others).

Note: If the baby was conceived with another man’s sperm, there is some issue of the mother’s husband making the berakhah of “lehachniso“, recited at the Brit (Rema YD 265:1). If this reality is known to the Rabbi, then ideally s/he should convene a beit din in private and appoint him as their representative to make the berakhah. However, even if this is not done, we can just assume that the kahal/virtual beit din wants him to represent them to make the berakhah.