I think it should be possible for the Java/Objective C code to save the start time as one of the first things it does and make it available to the JS code.
Maybe you could create a feature request. Of course if you are able to figure out how to do it in the Expo code maybe the Expo team would be willing to incorporate your change, which I suppose would probably be the fastest way to get this implemented.
I am not quite sure how autolinking fits in, but I don’t think it’s relevant to the Managed Worflow. The Expo team is apparently working on things like customising builds, which I think would help with things like this, but I don’t know how far away that is.
In the mean time, if you haven’t yet voted on this feature request which I’ve just found again and realised is what you’re looking for, I suggest you do so.