At this step you should have walked away and try another SSA office. The employees there are dumb. I'm 99% sure all the data was there, just this particular person had a bad day. My story - went to one office, they told me the same as to you: "Cannot see or check anything, file application to issue SSN, wait 2 weeks". I tried to go to another clerk in the same office, but he was sitting too close to the first one, so she noticed and screamed at me. Then I walked away, drove 15 minutes to another office and there was super friendly person who has checked in a few places in their system and in 5 minutes just wrote me my SSN on the sheet of paper. Received the physical mail a few days later btw, just needed to get the number itself. You can say something like "Need to start a job earlier" or so.
While there might be indeed the chance the data from DS260 wasn't sent/propagated, I see it as *very* unlikely event. So worth trying at least few different clerks and offices before you file an application.