A psak was just published which is supposedly similar to piskei halacha given by both Rav Moshe Shternbuch (of the Eida) and by Rav Elyashuiv..

the psak was writen by Rav Yehoshua Weissinger, of the Hesder Yeshiva Orot Yaakov, saying that there is a basis in halacha to postpone a bris to the 9th day in order to avoid chilul shabbos by family members who would drive to the bris on shabbos.

The common psak is that we do not push off the bris from Shabbos, for any reason (besides for reasons like a c-section or an issue of the exact timing of the birth), but there are opinions that say that if there is going to be definite chilul shabbos because of the bris, then it is better to push it off. Rav Wosner is quoted as being a source for postponing the bris in such a situation.

Rav Weissinger also points to rav Shternbuch who said that if the father himself will be involved in chilul shabbos then the bris should be postponed. Rav Elyashiv as well is quoted as having said that if the parents are mechalel shabbos, and will be for the bris, it should be postponed to Sunday.

Rav Weissinger adds that in some places, such as Petach Tikva, in order to avoid the problem, all circumcisions in those places, both for families that keep Shabbos and for families that do not, on Shabbos should only take place in the house of the family, rather than in a social hall or in the shul.
source: Charedim10 News

I would note that nothing is 100%. Even the rule given that in some places all circumcisions are in the home, is not completely true. I dont know if such a rule exists or not, but I asked someone in Petach Tikva and she told me that just yesterday on Shabbos she went to the shul for a bris... so I dont know how widely accepted this is