Someone in the government certainly knows more about Obama than has been made public. My trust in the US government is just slightly higher than my trust in Putin. Neither discloses anything but self serving info, and whether even what is released is true or not is a matter of speculation.
Trump absolutely has no access to Cruz’ Consular File. He knows no more about the contents than I do, which is squat. And that is by design.
Given that Obama has already opined that Cruz is worse than Trump, I would expect Obama to be spilling any information under his control which he regards as derogatory towards Cruz.
That Obama has not done so implies strongly that there isn't anything he can lay his grubby little fingers on.