“hat is, if we have some allowing, others – not, would carrying be always considered hillul shabbos by those who do not allow, or it maybe something like “ok b’dieved” or “ok when going for a mitzva” or “ok when not observed by public”?”
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.
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.