If you have everything installed now on a single drive / same bb folder should be fairly easy. I would attempt this:
On new computer, just install latest core version of BIAB software you purchased. Copy bb folder from your old computer to new, skipping identical files. (There should be an option "do this for all files"). Activate BIAB on new machine. Apply latest patch.

A better way would be to install everything fresh, but ^ should in theory work. Perhaps someone will correct me.