I've done a fair bit of research into Oppenheimer (not so much Barbie). He was most certainly a member of the Cabal. Though the entire Manhattan project was directly funded and managed by Cabal agents, 100%.

I think it is likely that every single person who worked on that project was similarly affiliated, though it is impossible to know exactly what level of "aware" anyone is. But when it comes to Oppenheimer, I am fairly certain he was completely aware of who he was working for.