I think that's right. The opening bay triggered a warning & O'Brien entered in response. The bridge could've known too, but no one was able to show up as quicklyYou're right about the monitor in the shuttlebay. Maybe that's why O'Brien mysteriously went in there when we hadn't seen him for the entire episode, or maybe he was alerted when the bay doors were opened. He might have gotten an alert and went to check on it. But, I guess no one on the Bridge knew about it. They usually get an alert when the bay doors are opened.
I doubt it'd be tough to differentiate Picards at that point. The foreign one was intent on leaving, & it's likely to have been recorded inside the shuttlebay. I bet it was weird for the computer locator though, given both had the same signature & com signal. They might've even been monitoring things via the coms too... maybe, even though that's never said