Damian huffed but he didn't outright protest. If anything, S.T.A.R. Labs would be a useful starting point in unraveling the details of the situation. It wasn't the Batcomputer or the databanks at the Justice League H.Q. but it was a start.
It would also give him a sense of the world he was dealing with. The process should following the examination should lead to shared knowledge as well as temporary accommodations. Anything else would be dealt with as needed. It was a start and, given the current state of affairs, not one he was in a position to protest.