Neville Chaim Berlin,
“Depends on the machlokes, and it’s really a question for your rav. Even by those who hold Brooklyn is a Reshus Harabim (the mainstream yeshivish opinion), those who carry in the eruv probably don’t get the status of not being shomer shabbos since they’ve been “mislead” into believing it’s ok. That’s usually how we get around these things, but on a theoretical level, yes, they are being machalel shabbos according to the machmirim.”
Actually Rav Moshe wrote (IM, O.C., 1:186) that when follows one’s rav on any issue, even on issurei chilul Shabbos, albeit the halachah is not like their rav’s interpretation, no aveirah is transgressed. Hence, they are not classified as mechalal Shabbos.
“As far as the “ok when going for a mitzvah,” no. There is no heter to be over an isser d’oraisa on Shabbos for a mitzvah. This is exactly why we don’t blow shofar or take arba minim on Shabbos.”
Actually, as most poskim maintain that there is no reshus harabbim today, they would need to grapple with this issue. In fact, many poskim maintain that the reason we are machmir today regarding shofer and arba minim on Shabbos is because of the original takanah (see Maharsham, OC, 8:25). While the reason is not nogiea anymore, we are not mevatal the gezeirah (there are other reasons as well).